Post on 04-Apr-2018
NONLINEAR STOCHASTIC MODELING AND
ANALYSIS OF CARDIOVASCULAR SYSTEM
DYNAMICS – DIAGNOSTIC AND PROGNOSTIC
APPLICATIONS
By
HUI YANG
Bachelor of Science China University of Mining and Technology
Beijing, China, 2002
Master of Science China University of Mining and Technology
Beijing, China, 2005
Submitted to the Faculty of the Graduate College of the
Oklahoma State University in partial fulfillment of
the requirements for the Degree of
DOCTOR OF PHILOSOPHY May, 2009
ii
NONLINEAR STOCHASTIC MODELING AND
ANALYSIS OF CARDIOVASCULAR SYSTEM
DYNAMICS – DIAGNOSTIC AND PROGNOSTIC
APPLICATIONS
Dissertation Approved:
Dr. Satish T. S. Bukkapatnam
Dissertation Adviser
Dr. Ranga Komanduri Co-Adviser
Dr. William J. Kolarik
Dr. Zhenyu (James) Kong
Dr. A. Gordon Emslie Dean of the Graduate College
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COPYRIGHT
BY
HUI YANG
December, 2008
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ACKNOWLEDGMENTS
I would like to express my deepest appreciation to my academic advisor, Dr.
Satish T. S. Bukkapatnam, for his academic advice, inspirations, continuous motivations
and generous financial support for my Ph. D study. I also owe my infinite gratitude to my
co-advisor, Dr. Ranga Komanduri for his guidance, high standard, professionalism and
immense impact on my research. Dr. Satish and Dr. Komanduri are wonderful teachers,
excellent researchers, and inspiring advisors – the perfect mentors every graduate student
can dream of. I feel fortune to come across them in my life. Sincere thanks are also due to
my dissertation committee member, Dr. William J. Kolarik for introducing me heart
reliability, risk analysis concepts. Special thanks to my dissertation committee member,
Dr. James Kong for teaching me Bayesian statistics and providing technical advice. My
heartfelt thanks are also owed to Dr. Bruce Benjamin, Dr. Martin Hagan, Dr. Lionel Raff
and Dr. Randy S. Wymore, from Center of Signal Analysis and Integrated Diagnostics
(SAID), for introducing neural network design, cardiovascular physiological knowledge
and ECG feature extraction techniques.
Many thanks to all the friends and colleagues for their great friendship and
encouragement throughout my PhD study in Stillwater.
My extreme appreciation goes to my wife, Yingfei Jia for her love, understanding
and support at the time of difficulties.
Finally, thank my beloved parents for their encouragements, sacrifices, patiences
and affections! I dedicate my dissertation to my dear father and mother!
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TABLE OF CONTENTS
CHAPTER I ...................................................................................................................... 1�
INTRODUCTION............................................................................................................. 1�
1.1� Research motivations ........................................................................................... 1�
1.2� Research objectives .............................................................................................. 3�
1.3� Major contributions .............................................................................................. 4�
1.4� Organization of the dissertation ........................................................................... 5�
1.5� References ............................................................................................................ 7�
CHAPTER II ..................................................................................................................... 8�
BACKGROUND AND LITERATURE REVIEW ........................................................ 8�
2.1� Cardiac conduction system ................................................................................... 8�
2.2� 12-Lead electrocardiogram (ECG) and vectorcardiogram (VCG) signals ........... 9�
2.3� A primer on nonlinear dynamic analysis ............................................................ 12�
2.4� Wavelet analysis of ECG signals ....................................................................... 17�
2.5� Databases ............................................................................................................ 22�
2.6� References .......................................................................................................... 24�
CHAPTER III ................................................................................................................. 29�
RESEARCH METHODOLOGY .................................................................................. 29�
3.1� Signal representation .......................................................................................... 30�
3.1.1 Nonlinear adaptive wavelet representation of ECG signals ............................ 30�
3.1.2 Dynamic VCG representation .......................................................................... 31�
3.2� Feature extraction and classification .................................................................. 32�
3.2.1 Atrial fibrillation state classification ................................................................ 32�
3.2.2 Myocardial infarction identification using recurrence quantification analysis of VCG signals .............................................................................................................. 32�
3.2.3 Vectorcardiographic octant features for the diagnostics of heterogeneous myocardial infarction ................................................................................................ 33�
3.3� Simulation and prognostic modeling.................................................................. 35�
3.3.1 Local recurrence prediction in nonstationary chaotic systems ........................ 35�
3.3.2 Radial basis function simulation modeling of VCG signals ............................ 36�
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PART I SIGNAL REPRESENTATION
CHAPTER IV.................................................................................................................. 37�
CUSTOMIZED WAVELET REPRESENTATION OF ECG SIGNALS ................. 37�
4.1� Background ........................................................................................................ 38�
4.2� Phase space fiducial pattern extraction .............................................................. 40�
4.3� Least square matching wavelet design ............................................................... 43�
4.4� Implementation and results ................................................................................ 46�
4.5� Summary ............................................................................................................ 53�
4.6� References .......................................................................................................... 54�
CHAPTER V .................................................................................................................... 57�
DYNAMIC VCG REPRESENTATION ........................................................................ 57�
5.1� Background ......................................................................................................... 58�
5.2� Research methodology ........................................................................................ 60�
5.2.1� Spatiotemporal VCG attractor representation .............................................. 61�
5.2.2� Lag reconstructed spatiotemporal ECG attractor ......................................... 64�
5.2.3� Color coding of spatiotemporal VCG representation .................................. 68�
5.3� Results and discussion ......................................................................................... 70�
5.4� Concluding remarks ............................................................................................ 79�
5.5� References ........................................................................................................... 80�
PART II FEATURE EXTRACTION AND CLASSIFICATION CHAPTER VI.................................................................................................................. 83 ATRIAL FIBRILLATION STATE CLASSIFICATION ........................................... 83
6.1 Introduction ........................................................................................................ 83 6.2 Background ........................................................................................................ 84 6.3 Feature extraction ............................................................................................... 87
6.3.1 Statistical Descriptors and Principal Component Analysis (PCA) .................. 87 6.3.2 QRST subtraction............................................................................................. 90
6.4 AF state classification using CART model ........................................................ 92 6.5 Summary ............................................................................................................ 98 6.6 References .......................................................................................................... 99
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CHAPTER VII .............................................................................................................. 105 MYOCARDIAL INFARCTION IDENTIFICATION USING RECURRENCE QUANTIFICATION ANALYSIS (RQA) OF VCG SIGNALS ................................ 105
7.1 Introduction ...................................................................................................... 105 7.2 Research methodology ..................................................................................... 107 7.3 Implementation and results .............................................................................. 112
7.3.1 Recurrence quantification and feature extraction ..................................... 113 7.3.2 Feature analysis ......................................................................................... 115 7.3.3 Classification............................................................................................. 120
7.4 Summary .......................................................................................................... 124 7.5 References ........................................................................................................ 126
CHAPTER VIII ............................................................................................................ 130 VECTORCARDIOGRAPHIC OCTANT FEATURES FOR THE DIAGNOSTICS OF HETEROGENEOUS MYOCARDIAL INFARCTION ..................................... 130
8.1 Introduction ...................................................................................................... 130 8.2 Feature extraction ............................................................................................. 132 8.3 Feature analysis ................................................................................................ 137 8.4 Classification .................................................................................................... 139 8.5 Discussions ....................................................................................................... 144 8.6 Summary .......................................................................................................... 146 8.7 References ........................................................................................................ 147
PART III SIMULATION AND PROGNOSTIC MODELING
CHAPTER IX................................................................................................................ 150 LOCAL RECURRENCE PREDICTION IN NONSTATIONARY CHAOTIC SYSTEMS ...................................................................................................................... 150
9.1 Introduction ...................................................................................................... 150 9.2 Background ...................................................................................................... 153 9.3 Local attractor topology analysis ..................................................................... 156 9.4 Simulation experiment results .......................................................................... 163 9.5 Summary .......................................................................................................... 169 9.6 References ........................................................................................................ 170
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CHAPTER X ................................................................................................................. 172 RADIAL BASIS FUNCTION SIMULATION MODELING OF VCG SIGNALS 172
10.1 Poincare sectioning of vectorcardiogram ......................................................... 173 10.2 Heart rate variability ......................................................................................... 174 10.3 Research methodology ..................................................................................... 176 10.4 Transformation from VCG to standard 12 lead ECG signals .......................... 178 10.5 Implementation results ..................................................................................... 178 10.6 Summary .......................................................................................................... 185 10.7 Reference .......................................................................................................... 186
CHAPTER XI................................................................................................................ 189 CONCLUSIONS AND FUTURE WORK .................................................................. 189
11.1 Conclusions ...................................................................................................... 189 11.2 Future work ...................................................................................................... 191
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LIST OF TABLES
Table 6.1 Distribution of 80 recordings among learning sets and test sets ....................... 85
Table 6.2 Summary of final results of the 2004 PhysioNet Challenge ............................. 86
Table 6.3 S vs T learning data set RR interval and skewness of RR feature table ........... 96
Table 6.4 S vs T testing data set RR interval and skewness of RR feature table ............. 96
Table 7.1 Linear model misclassification results using F1 and other principal features (F2
F6) ................................................................................................................................... 116�
Table 7.2 Relative importance of principal features F1 to F6 using linear regression ... 121�
Table 7.3 Results of linear regression classification for MI and HC .............................. 121�
Table 7.4 Results of neural network (NN) classification for MI and HC ....................... 123�
Table 8.1 VCG octant definition and color mapping ...................................................... 135
Table 8.2 Statistical analysis (Kolmogorov-Smirnov test) of three groups of features .. 138
Table 8.3 Experiment results of CART classification model using stochastic training
datasets ............................................................................................................................ 140
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LIST OF FIGURES
Figure 1.1 Pie chart of death causes in the world .............................................................. 1�
Figure 2.1 Electrophysiology of the heart and different waveforms for each of the
specialized cells in the heart ............................................................................................... 8�
Figure 2.2 Electrical and mechanical events diagram during one heart beat ...................... 9�
Figure 2.3 The electrodes placement in VCG measurement system ................................ 10�
Figure 2.4 Rossler attractor and local evolution ............................................................... 13�
Figure 2.5 Rossler attractor recurrence plot ...................................................................... 15�
Figure 2.6 Time-frequency resolution of wavelet transforms .......................................... 18�
Figure 3.1 Schematic of research methodology ................................................................ 29
Figure 4.1 Procedure for customized wavelet design from system nonlinear dynamics .. 38�
Figure 4.2 An illustration of trajectories of an attractor intersecting a (planar) Poincare
section ............................................................................................................................... 41�
Figure 4.3 Time domain plot of a representative ECG signal trace (a01) ........................ 47�
Figure 4.4 State space portrait reconstructed from time delay coordinates ...................... 48�
Figure 4.5 ECG pattern ensembles extracted from Poincare section ................................ 49�
Figure 4.6 Matching wavelet extracted from fiducial signal pattern ................................ 49�
Figure 4.7 Matching wavelet representation entropy variation with various polynomial
degrees .............................................................................................................................. 50�
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Figure 4.8 Entropy comparison between standard wavelets and matching wavelets ....... 51�
Figure 4.9 Entropy distribution for different wavelet representations, taken over all ECG
signals in the database considered .................................................................................... 52�
Figure 4.10 Histogram of matching wavelet transform coefficients: (a) customized
wavelet, (b) Morlet wavelet .............................................................................................. 53�
Figure 5.1 Frames from dynamic VCG representation of a patient with myocardial
infarction, anterior .............................................................................................................. 63
Figure 5.2 Frames from lag reconstructed dynamic EKG attractor representation for a
normal subject .................................................................................................................... 67
Figure 5.3 Color coded dynamic VCG representation plot of a healthy control subject ... 69
Figure 5.4 Color coded dynamic VCG representation plot of a patient with dysrhythmia
and atrial fibrillation .......................................................................................................... 72
Figure 5.5 Color coded dynamic VCG representation plot of a patient with myocardial
infarction, infero-latera ...................................................................................................... 74
Figure 5.6 Color coded dynamic VCG representation plot of a patient with myocardial
infarction, antero-lateral ..................................................................................................... 76
Figure 5.7 Color coded dynamic VCG representation plot of a patient with bundle branch
block ................................................................................................................................... 78
Figure 6.1 Automated identification of peaks and endpoints of various waveforms in an
ECG signal using an automated feature extraction routine .............................................. 88
Figure 6.2 Time-scale representation (scalogram) of a representative ECG signal ......... 89
Figure 6.3 Summary of QRST subtraction results for a representative ECG signal (from
a T-case) showing the original signal (top), filtrate from thresholding containing
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predominantly ventricular components (middle) and residue from thresholding containing
predominantly atrial and AF waves (bottom) ................................................................... 91
Figure 6.4 Frequency spectra of a representative ECG signal before and after QRST
subtraction ......................................................................................................................... 92
Figure 6.5 (a) Peak frequencies of various N and T cases, (b) Peak frequencies of the first
two (First 2) and last two (Last 2) beats of N and T ECG signals .................................... 93
Figure 6.6 (a) and (b) Learning and testing dataset RR interval and average features
scatter plot, respectively. (Average feature is the average value of mean of QRS, mean of
peak, mean of height of T wave, mean of QT interval, mean of ST segment, and mean of
T duration) ........................................................................................................................ 95
Figure 7.1 Coordinate system used for obtaining VCG signals...................................... 107�
Figure 7.2 A representative VCG plot showing the vector loops for P, QRS, and T wave
activities .......................................................................................................................... 108�
Figure 7.3 A graphical illustration of relationship among a VCG, the X,Y,Z time series
and the unthresholded recurrence plot ............................................................................ 110�
Figure 7.4 A representative thresholded recurrence plot of VCG .................................. 111�
Figure 7.5 Flow diagram showing research methodology used ..................................... 112�
Figure 7.6 Histogram comparisons between HC (filled green) and MI(not shaded) for the
six RQA quantifiers, namely, recurrence rate (�), determinism (DET), linemax (LMAX),
entropy (ENT), laminarity (LAM), and trapping time (TT) (Histogram x-axis is feature
value, and y-axis is the normalized frequency) .............................................................. 114�
Figure 7.7 Histogram distributions of PCA feature values F1 to F6 for MI (not shaded)
and HC (shaded green) (Histogram x-axis is feature value and y-axis is the normalized
frequency) ....................................................................................................................... 117�
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Figure 7.8 (a) to (e) Distribution of values of features F2 through F6 vs. F1 for MI (red �)
and HC (blue +) subjects with linear separator obtained using a genetic algorithm ...... 118�
Figure 7.9 Relative importance of RQA quantifiers in the first principal feature (F1) .. 119�
Figure 7.10 Radial basis function (RBF) neural network (NN) ...................................... 122�
Figure 7.11 Variation of % accuracy of the neural network (NN) (both training and
testing) with the number of data points for MI and HC .................................................. 124�
Figure 8.1 Frank XYZ VCG ensembles, (a) X-direction; (b) Y-direction, (c) Z-direction
......................................................................................................................................... 132
Figure 8.2 VCG attractor and its 2D projection plots with Q, R, T vectors ................... 133
Figure 8.3 (a) VCG electrodes placement in human body; (b) the corresponding octant
definitions ....................................................................................................................... 134
Figure 8.4 VCG trajectories with different octant colors in the 3D coordinate system, (a)
HC; (b) MI ...................................................................................................................... 136
Figure 8.5 Kolmogorov-Smirnov statistic variations of 46 features ............................... 137
Figure 8.6 CART model cross validation curves and Tree size selection ...................... 141
Figure 8.7 Optimal CART tree structure for the classification of MI in PTB database . 142
Figure 8.8 Detailed classification of PTB database MI and HC in (a) step 1 and (b) step 2
......................................................................................................................................... 143
Figure 8.9 Healthy control (368) and myocardial infarction (80) distribution in 3D feature
space (Oct0MaxN, T octant position and Oct2AvgN) .................................................... 144
Figure 8.10 Oct0MaxN (a), T octant position (b) and Oct2AvgN (c) histogram plots .. 145
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Figure 9.1 Graphical illustration of the relationship among Lorenz time series, attractor
and recurrence plot .......................................................................................................... 153
Figure 9.2 Overall framework of local recurrence prediction model ............................. 157
Figure 9.3 Edge detection in a Lorenz recurrence plot ................................................... 160
Figure 9.4 Local eigen prediction model from ensembles within a stationary segment . 161
Figure 9.5 (a) top - original Lorenz time series, middle – random Gaussian noise, bottom
– random permutated Lorenz time series contaminated by noise; (b) reconstructed noise-
free Lorenz attractor vs. random permutated Lorenz attractor contaminated by noise .. 164
Figure 9.6 Five step prediction RMS error comparisons for Lorenz time series under
different noise levels with various prediction models (ARMA(1,0), ARMA(3,3), RBF
model and local recurrence model) ................................................................................. 166
Figure 9.7 Five step prediction RMS error comparisons for randomly permutated Lorenz
time series under different noise levels with various prediction models (ARMA(1,0),
ARMA(3,3), RBF model and local recurrence model) ................................................... 168
Figure 10.1 Overall framework of RBF simulation model of VCG signals ................... 172
Figure 10.2 (a) An illustration of trajectories of an attractor intersecting a (planar)
Poincare section; (b) Poincare sectioning of VCG trajectory ......................................... 173
Figure 10.3 X, Y and Z ensemble components extracted from the Poincare sectioning of
VCG trajectory ................................................................................................................ 174
Figure 10.4 Weibull distribution fitting of RR interval time series ................................ 175
Figure 10.5 Uniform distributed radial basis function centers along VCG trajectory .... 176
Figure 10.6 Patient001/s0010_re (a)Simulated VCG trajectory vs. actual VCG trajectory;
(b) Frequency domain comparison between actual VCG and simulated VCG; (c) Time
domain comparison between actual VCG and simulated VCG ...................................... 180
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Figure 10.7 Patient255/s0491_re (a)Simulated VCG trajectory vs. actual VCG trajectory;
(b) Frequency domain comparison between actual VCG and simulated VCG; (c) Time
domain comparison between actual VCG and simulated VCG ...................................... 181
Figure 10.8 Patient005/s0101lre (a)Simulated VCG trajectory vs. actual VCG trajectory;
(b) Frequency domain comparison between actual VCG and simulated VCG; (c) Time
domain comparison between actual VCG and simulated VCG ...................................... 182
Figure 10.9 Patient106/s0030_re (a)Simulated VCG trajectory vs. actual VCG trajectory;
(b) Frequency domain comparison between actual VCG and simulated VCG; (c) Time
domain comparison between actual VCG and simulated VCG ...................................... 183
Figure 10.10 Patient175/s0009_re (a)Simulated VCG trajectory vs. actual VCG
trajectory; (b) Frequency domain comparison between actual VCG and simulated VCG;
(c) Time domain comparison between actual VCG and simulated VCG ....................... 184
1
CHAPTER I
INTRODUCTION
In this chapter, motivations for selecting this research topic are presented. It is
followed by the major thrust and scope of research. In the end, the organization of this
dissertation is shown.
1.1 Research motivations
Figure 1.1 Pie chart of death causes in the world [1]
According to the World Health Organization (WHO) reports in 2008 (see Figure 1.1),
cardiovascular disease is the number one cause of death globally and is projected to
remain the leading cause of death. An estimated 17.5 million people died from
cardiovascular disease in 2005, representing 30% of all global deaths. Of these deaths,
7.6 million were due to heart attacks and 5.7 million were due to stroke [1].
2
There are significant socioeconomic values for the diagnosis and prognosis of
cardiovascular diseases. It is estimated that if appropriate action is taken by 2015, twenty
million people will be saved from cardiovascular diseases every year [1]. The
cardiovascular diseases will affect the quality of life of many people, increase the
healthcare costs, take away people’s earning and savings, and pose a heavy burden for the
country’s economies. Therefore, it is imperative to prevent and control the cardiovascular
diseases.
The major tool used in clinical diagnostics of cardiovascular disorders is
electrocardiogram (ECG), which is developed by Dr. William Einthoven in 1901. The
ECG is a time-varying signal that captures the ionic current flow responsible for the
contraction and subsequent relaxation of the cardiac fibers. The surface ECG is obtained
by recording the potential difference between two electrodes placed on the surface of the
skin.
Currently, with the advancement of information technology and sensor technology,
the acquisition of ECG signals is not constrained by computation resources any more.
The conventional 12-lead ECG and Frank XYZ VCG system enable the inspection of the
heart from different perspectives and the ECG data can be digitalized at a high sampling
rate and stored for many years. The availability of vast amounts of heart monitoring data
offers an unprecedented opportunity to develop accurate real-time diagnostics and
prognostics of cardiovascular system state and performance from a nonlinear dynamic
perspective.
3
1.2 Research objectives
The study is to develop real-time monitoring and prognostic schemes for various
cardiovascular diseases by analyzing the nonlinear stochastic dynamics underlying the
complex heart system.
Multianalysis strategies are used to diagnose the patient’s cardiovascular online status
and make a further prognosis to assess the risk of heart disease. The employment of a
nonlinear stochastic analysis combined with wavelet representations can extract effective
cardiovascular features for this purpose. Those features will be more sensitive to the
pathological dynamics instead of the extraneous noises. The conventional approaches
(statistical and linear system) tend to have limitations for capturing signal variations
resulting from changes in the cardiovascular system state.
The study includes signal representation, feature extraction, state classification and
state prognosis from nonlinear dynamic principles. In cope with the main research
objectives, the research tasks are divided into the following:
� Develop an effective compact representation method for the cardiovascular
signals which will yield sensitive features to relevant ECG states and insensitive
to variations in extraneous noise.
� Characterize the nonlinear spatiotemporal dynamics of cardiac vector loops and
quantifying the recurring patterns in the vectorcardiogram (VCG) for the
Myocardial Infarction (MI) classification.
4
� Develop the classification and regression tree (CART) model to integrate the
medical doctor’s knowledge towards the classification of three Atrial Fibrillation
(AF) states
� Develop the real time prediction model for the heart risk assessment under
dynamic conditions.
� Design the data-driven simulation model of cardiac electrical activities in various
cardiovascular conditions.
1.3 Major contributions
This proposed research will provide versatile cardiovascular monitoring and
prognosis schemes which combine the nonlinear dynamics, predictive modeling, artificial
intelligence, and signal processing. Specific research contributions are as follows:
� New representation scheme for the ECG signals by an optimal wavelet function
design which yields one order of magnitude compactness reduction compared to
the conventional wavelets in terms of entropy.
� New spatiotemporal representation scheme for the VCG signals, which
significantly assists the interpretation of vectorcardiogram and facilitates the
exploration of related important spatiotemporal cardiovascular dynamics.
� Developed a classification and regression tree (CART) model to integrate the
cardiologists’ knowledge towards accurate classification of Atrial Fibrillation
(AF) states from sparse datasets.
5
� Developed a Myocardial Infarction (MI) classification model using recurrence
quantification analysis and spatial octant features of VCG attractor dynamics,
which detects MI up to about 96% accuracy in both sensitivity and specificity.
� Designed a local recurrence prediction model for real time medical prognosis and
risk assessment of cardiovascular problems, which is particularly suitable for the
implementation in the nonlinear and nonstationary environment.
1.4 Organization of the dissertation
This chapter presents the motivation of research in this area, as well as research
objectives and research contributions. Rest of the dissertation is organized as follows:
Chapter II: Background and literature review: This chapter describes the general
fundamentals of cardiovascular electrical activities and relevant monitoring systems.
Next, the databases used in the subsequent chapters for experimental and simulation
studies are introduced. In addition, a review of pertinent literature with the wavelet,
nonlinear dynamic study of ECG signals is presented.
Chapter III: Research methodology: This chapter presents the research
methodology used in the dissertation study. The overall research methodology is grouped
into three parts: signal representation, feature extraction and classification, and simulation
and prognostic modeling.
PART I: Signal representation
Chapter IV: Customized wavelet representation of ECG signals: An optimal
wavelet function design approach is established for the specific ECG signal
representation. Implementation details, validations, results and conclusions are described.
6
Chapter V: Dynamic VCG representation: The time resolution drawbacks of
conventional static VCG representation are resolved by real-time displaying the rotated
VCG attractor on a computer monitor. It explores the VCG spatiotemporal information
on the heart dynamics. The color coding methods of VCG attractor using cardiac vector
curvature, velocity, and phase angle information are also introduced.
PART II: Feature extraction and classification
Chapter VI: Atrial fibrillation state classification: The approach, development,
training, validation of atrial fibrillation classification and wavelet feature extraction are
presented. The classification and regression tree (CART) model is detailed.
Chapter VII: Myocardial infarction identification using recurrence
quantification analysis (RQA) of VCG signals: The recurrence behaviors and
properties of VCG attractor are characterized and quantified. Recurrence quantifiers are
used for the identification of myocardial infarction cases from the healthy control objects.
Chapter VIII: Vectorcardiographic octant features for the diagnostics of
heterogeneous myocardial infarction: The spatial distributions of VCG trajectories are
designed and studied in this chapter. The critical VCG octant features are extracted for
the diagnostics of heterogeneous MIs. The training, validating, testing of CART
classification models are detailed.
PART III: Simulation and prognostic modeling
Chapter IX: Local recurrence prediction in nonstationary chaotic systems: A
local recurrence modeling approach is presented for the state and performance
predictions in complex nonlinear and nonstationary systems. Extensive studies using
7
simulated and some real world datasets are shown to achieve significant prediction
accuracy improvements over other alternative methods.
Chapter X: Radial basis function simulation modeling of VCG signals: The
parameterized RBF network model is studied for the generation of simulated VCG
signals and they are also transformed to get the standard 12-lead EKG signals by the
generalized Dower transformation matrix or the estimated matrix from experiment data
for the specific patient.
Chapter XI: Conclusions and future work: This chapter presents research
contributions, general conclusions and future work.
1.5 References
[1] "http://www.who.int/cardiovascular_diseases/en/," World Health Organization,
2008.
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CHAPTER II
BACKGROUND AND LITERATURE REVIEW
This chapter will begin with the literature review about cardiac conduction system,
12-lead ECG and frank XYZ VCG system. It will be followed by the relevant research on
the nonlinear dynamic and wavelet analysis of ECG signals.
2.1 Cardiac conduction system
Figure 2.1 Electrophysiology of the heart and different waveforms for each of the
specialized cells in the heart [1]
The heart consists of four compartments: the right and left atria and ventricles. It
works as an autonomous control pump for circulating the blood in our bodies and
constantly produces a sequence of electrical activities within every heart beat. The heart
9
electrical activities are generated by the biopotential variations because of the movement
of three types of ions, namely, Calcium (Ca++), Potassium (K+) and Sodium (Na+). As
shown in Figure 2.1, the heart's electrical activity begins in the sinoatrial (SA) node to
excite the atrial muscle contraction. Then AV node will propagate the stimulus through
bundle of His and Purkinje fibers toward the ventricles. The SA node is the heart's
pacemaker. The ordered stimulation starting from SA node leads to the efficient
contraction of the heart, thereby pumping the blood throughout the body [2].
2.2 12-Lead electrocardiogram (ECG) and vectorcardiogram (VCG) signals
Figure 2.2 Electrical and mechanical events diagram during one heart beat [3]
Cardiac monitoring system is designed to measure the electrical biopotential
changes in the heart. The electrocardiogram (ECG) system, designed by Augustus Waller
in 1889 and further improved by Williem Einthoven in 1901, has been used for more than
100 years for the clinical diagnosis of cardiovascular disorders. Around 1904 Einthoven
derived the famous “Einthoven triangle” to measure three ECG signals and calculate the
approximate direction of cardiac vector. In 1956, Ernest Frank designed the Frank lead
10
system to measure the vectorcariogram (VCG) in three corrected orthogonal coordinates,
which provides a clear picture of the cardiac vector.
The surface ECG is obtained by recording the potential difference between two
electrodes placed on the surface of the skin. A single normal cycle of the ECG represents
the successive atrial and ventricular depolarization-repolarization events which occur
with every heart beat and includes P, Q, R, S, T, and U waves (see Figure 2.2)
� P wave: the sequential activation of the right and left atria
� QRS complexes: right and left ventricular depolarization
� T wave: ventricular repolarization
� U wave: origin though not clear, is probably “post-depolarization” in the
ventricles
Figure 2.3 The electrodes placement in VCG measurement system [1]
11
The vectorcardiogram (VCG) observes the heart potentials as a cardiac vector in
three orthogonal components instead of the scalar amplitude (ECG curve). In the
rectangular coordinate system of the torso, VCGs are the mutually orthogonal bipolar
measurements by placing parallel electrodes on the opposite sides of the torso (see Figure
2.3).
With the rapid development of information technology, the representation and
analysis of 3D VCG loops are not constrained by computational resources and this
resulted in renewed interests of VCG since 1990’s. As demonstrated by Dower and his
group [4-6], VCG signals and 12-lead ECG can be linearly transformed to each other
without loss of useful information content pertaining to the heart dynamics. Dower
matrix (Eq 2-1) is one of those generalized transformation matrices, and can be used to
derive the 12-lead ECG signals from the 3-lead VCG.
Dower transformation matrix
0.515 0.157 0.9170.044 0.164 1.3870.882 0.098 1.2771.213 0.127 0.6011.125 0.127 0.0860.831 0.076 0.2300.632 0.235 0.0590.235 1.066 0.132
D
� �� �� ��� �� ��� ��� � � ��� �� �� ��� �
�� � �
(Eq 2-1)
If we have the VCG data in matrix form V (3×N), the eight leads (I, II, v1-v6) S (8×N)
can be derived by S=DV except the derived augment leads.
The transformation study statistically shows the equivalence between 12-lead ECG
and 3-lead VCG. However, it is difficult for the human beings to visually project a spatial
VCG vector into any specified cardiac measurement angle determined by 12-lead
12
measurement systems, which is the conventional way to interpret heart electrical
activities. In addition, the loss of temporal information in the static VCG representation
posed extra barriers for medical doctors to closely relate the 12-lead ECG characteristics
to VCG. But when it comes to the computer automated analysis of heart monitoring
signals, the 12-lead ECG signals will introduce the “curse of dimensionality” problem,
which is the fact that the convergence of any estimator to the true value of a smooth
function defined on a space of high dimension is very slow because the exponential
increase in volume associated with extra dimensions. Thus, VCG is a better option for the
computer processing and analysis because it overcomes not only the loss of information
by only analyzing one or two ECG signals, but also the dimensionality problems induced
by 12-lead ECG. The usage of VCG for the diagnostics of cardiovascular diseases has
been studied by many researchers [7-18], although very few investigations explored both
spatial and temporal relationships.
2.3 A primer on nonlinear dynamic analysis
Much of the complexity in complex systems emerges from the underlying nonlinear
stochastic dynamics. Effective models of the following form (Eq 2-2) can capture this
complexity:
( ) ( ) � d F dt dx x x (Eq 2-2)
where x(t) is a m-dimensional state vector, F(•) is usually a nonlinear vector field, t is the
time and the dynamic noise term (x)d� accounts for the influence of extraneous
phenomena [19-23]. Nonlinear dynamic cardiovascular signal analysis is based on the
premise that the ECG signals emanate from a finite dimensional nonlinear dynamic
13
system. This is reasonable and more generic than conventional linear and stationary
assumptions on the nature of cardiovascular system and it can lead to effective models
that can compactly capture the salient behaviors of many complex systems. Dynamics
underlying the ECG signals gathered under these conditions manifest in the vicinity of an
invariant set called an attractor A, defined in the state space [24]. An example of such an
attractor of a nonlinear system (Rossler system) is shown in Figure 2.4. The state space of
Rossler and such nonlinear systems shows deterministic chaos where the state evolution
switches between a horn (emerging through the neighborhood U) and cycle area
(emerging through V).
Figure 2.4 Rossler attractor and local evolution
However, in many real life cases, only a few signals (e.g., one lead ECG signal) are
available in lieu of the complete measurements of the state vector x(t). However, many
individual signal y(t) contain adequate information to reconstruct most of the system
dynamics. This is because of the high dynamic coupling existing among cardiovascular
system. An equivalent state space (attractor) can be reconstructed from the delayed
coordinates of the measurements y(t) as
x(ti) =[y(ti), y(ti+�), y(ti+2�), y(ti+(m-1)�)] (Eq 2-3)
where m is the embedded dimension and � is the time delay [25, 26]. The minimal
-10
0
10
-15-10-505100
10
20
30
40
V
U
14
sufficient embedding dimension m to unfold the attractor is determined by false nearest
neighbor method [27]. The optimal time delay � is selected to minimize mutual
information function M(�), defined as
( , )( ) ( , ) log( ) ( )
�� ��
�t
p t tM p t t dtp t p t
(Eq 2-4)
where p(t,�) is the joint density function, and p(t) and p(t+�) are marginal density
functions of y(t) and y(t+�), respectively.
Recurrence plots provide a convenient means to capture the topological
relationships existing in the m-dimensional state space by calculating the distance from
each state to all the others ( , ) : ( ) ( ) �D i j v i v j , where ||·|| is a norm (e.g., the Euclidean
norm) and mapping the distance to a color scale. It shows the times at which a state of the
dynamical system recurs, i.e., the time-pairs at which the trajectories of a system
evolution come within a specified neighborhood. Recurrence plot is formed by
calculating the Euclidean distance from each state vector to others and mapping the
distance to a color scale. A representative recurrence plot constructed for the Rossler
attractor is shown in Figure 2.5. The structures of a recurrence plot have distinct topology
and texture patterns. The ridges locate the transition states at which the system evolves
from one behavior (cycle area V) to the other (horn area U) (see Figure 2.4). The
separations between dark diagonal lines (along 45o degree) indicate the near-periodicity
of system behaviors over a given time segments. Recurrence-based methods have of late
shown potential for representation and de-noising of measurements from complex
systems [28-34].
15
Figure 2.5 Rossler attractor recurrence plot
Recurrence plots are intriguing graphical displays of signal patterns, but
representative nonlinear dynamical quantitative features need to be extracted to analyze
the underlying processes and detect hidden rhythms. Statistical features, such as
recurrence rate (%RR), determinism (%DET), laminarity (%LAM), linemax (LMAX),
entropy (ENT), and trapping time (TT), are used as such typical recurrence quantifiers for
the complexity characterization of VCG attractor .
The definitions of various recurrence quantifiers and their relationships with heart
dynamics are summarized in the following [28-34]:
1) Recurrence rate (�) is the percentage of dark points on the TRP (the pairs of
points whose distance is less than the corridor) [28-34]. It is calculated as:
2
, 1
1 ( , )N
i jT i j
N
� � (Eq 2-5)
where N is window size and T(i,j) is the TRP with a value 1 or 0. Recurrence rate
characterizes the global near-periodicity of cardiovascular activities and it is
closely related to the heart rate dynamics. The higher the �, the larger the
proportion of points located close to each other, and fewer the points are farther
apart.
Time Index
Tim
e In
dex
500 1000 1500 2000
500
1000
1500
2000
10
20
30
40
16
2) Determinism (DET) represents the proportion of recurrent points forming
diagonal line structures. It measures the repeating or deterministic patterns in the
heart dynamics, and tells how well the circulatory heart system functions [28-34].
min
, 1
( )
( , )
N
l lN
i j
lp lDET
T i j
�
� (Eq 2-6)
where l represents the length of lines parallel to the central diagonal line, lmin
represents the minimal length of counted diagonal lines and p(l) is the frequency
distribution of diagonal line segments of length l. Periodic signal has long
diagonal lines (big DET�100%), while chaotic signal has short diagonal lines
(medium DET<50%) and a random white noise signal shows no diagonal lines
(DET � 0).
3) Linemax (LMAX) is the length of the longest diagonal line segment in the TRP,
which is related to the inverse of the largest positive lyapunov exponent [28-34].
The shorter the LMAX is, the more chaotic (less stable) is the signal. Thus,
LMAX indicates stability of heart dynamics, and small LMAX implies that two
close cardiac vectors in a VCG are diverging quickly from each other.
4) Entropy (ENT) is the Shannon information entropy of frequency distribution of
diagonal line segments [28-34]. The predictability of heart activity decreases with
increasing entropy given by
min
( ) ln ( )
��N
l lENT p l p l (Eq 2-7)
where p(l) is the frequency distribution of diagonal line segments of length l.
17
5) Laminarity (LAM) is analogous to DET except that it measures the vertical line
structures and is given by
min
1
( )
( )
�
�
N
v vN
v
vp vLAM
vp v (Eq 2-8)
where v represents the length of vertical lines, vmin represents the minimal length
of counted vertical lines and p(v) is the frequency distribution of vertical line
segments of length v. Large LAM indicates heart takes a significant time to move
from one activity to another, and it provides important non-stationary information
for the heart system [28-34].
6) Trapping time (TT) provides a measure of how long the system remains in a
specific state.
min
min
( )
( )
�
�
N
v vN
v v
vp vTT
p v (Eq 2-9)
These recurring pattern features are used to quantify the nonlinear dynamic
complexity of VCG vector loops under MI and HC conditions [28-34].
2.4 Wavelet analysis of ECG signals
ECG signals are gathered and stored in analytical instruments (e.g., ECG machines)
in the form of time-series. They can be transformed from the time domain into frequency,
time-frequency, or other domains depending on the nature of the information required.
These signals exhibit recurrent patterns spread over multiple scales, and significant
18
nonstationarity. Wavelet methods are most effective in capturing spatio-temporal content
of such signals over multiple scales of resolution.
Freq
uenc
y
Time
Figure 2.6 Time-frequency resolution of wavelet transforms
The wavelet representation can be useful for interrogating the spectral component
prevailing at a given time instant. The high frequency bursts, which usually occur over
short time intervals, can be analyzed with a sharper time resolution, while the low-
frequency components, which usually occur over long-duration of a signal, can be
analyzed with a better frequency resolution. Figure 2.6 shows various time and frequency
bands captured by different wavelets, i.e., each wavelet capture the time and frequency
range that from the boundaries of a particular box. Also, as illustrated in Figure 2.6,
despite the change of widths and heights of the boxes, the area of the boxes is constant.
At low frequencies, the height of the boxes are shorter which corresponds to better
frequency resolutions, but their widths are longer which correspond to poor time
resolution. At higher frequencies, the width of the boxes decreases, i.e., the time
resolution gets better and the heights of the boxes increase, i.e., the frequency resolution
gets poorer.
19
The admissible conditions for a real or complex-value continuous-time function
�(t) to be a “Wavelet” are as following:
(1) Reconstruction condition
2( )
0C d C�
��
�
��
� � � �� (Eq 2-10)
( )�� stands for the Fourier transform of �(t). This condition is sufficient, but not
a necessary condition to obtain the inverse. It can gurantee to reconstruct a signal without
loss of information. The admissibility condition implies that the Fourier transform of �(t)
vanishes at the zero frequency [35-37].
2
0( ) 0
��
� (Eq 2-11)
It indicates that wavelets must have a band-pass like spectrum.
(2) Wave condition
A zero at the zero frequency in condition (1) also means that the average value of
the wavelet in the time domain must be zero, which means �(t) must be an oscillatory
wave [35-37].
( ) 0t dt��
��
� (Eq 2-12)
(3) Finite energy
The signal to analyze must have finite energy. When the signal has infinite energy
it will be impossible to cover its frequency spectrum and its time duration with wavelets.
Usually this constraint is formally stated as
20
2( )t dt��
��
� �� 2 2( ) { : | | ( ) | }L R f R C f t dt�
��
� ��� (Eq 2-13)
and it is equivalent to saying that the L2-norm of our signal f(t) should be finite [35-37].
(4) Regularity condition
This condition is to make the wavelet transform decrease quickly with decreasing
scale s, and it states that the wavelet function should have some smoothness and
concentration in both time and frequency domains. Regularity is closely related to the
number of vanishing moments and it will determine how smooth a wavelet is. The order
of regularity of a wavelet is the number of its continuous derivatives. The j s moment of
the function �(t) is defined as ( )jt t dt� �
��� . When the wavelet s first k moments are zero
( ) 0 for 0,...,jt t dt j k� �
�� � , the number of Vanishing Moment of the wavelet is
k+1. If a wavelet has k vanishing moments, suppression of signals that are polynomials
of a degree lower or equal to k is ensured [35-37].
Summarizing, the admissibility conditions (1) (2) (3) gave us the wave, condition
(4) regularity and vanishing moments gave us the fast decay or the let, and a function
meeting the requirement of the above four conditions reaches the wavelet [38].
The Continuous Wavelet Transform (CWT) accomplishes the multi-resolution
analysis by time-scaling and time-shifting the wavelet function �(t) [35-37].
,1 t( )
jjj kk� � �
(Eq 2-14)
Where scaling factor j is for expanding or compressing and translation factor k is for the
21
time domain shifting. Continuous wavelet transform of the signal x(t) using the
analysis wavelet �(t) is defined as:
� �,
1j k
t
t kb x t dtjj
�� ��
� �� �
� (Eq 2-15)
Inverse continuous wavelet transform to reconstruct signal x(t) is:
, ,2
1 1( ) ( )j k j kj k
x t b t djdkC j
�� �
�� ��
�� � (Eq 2-15)
Actually, the continuous wavelet transform can be taken as ,( , ) ( ), ( )j kCWT j k x t t��
which is the cross correlation of the signal x(t) with the mother wavelet at scale j and lag
k. If x(t) is similar to the mother wavelet at this scale j and lag k, then the coefficients will
be large which indicates that customized wavelet function adapted to the ECG pattern
may achieve the best performance [35-37].
Wavelet functions �(t) are building blocks that can be used to simultaneously
decompose signal characteristics in both time and frequency domains. Wavelet
representations are particularly useful for the analysis of transients, aperiodicity, and
other nonstationary signal features. Subtle changes in signal morphology can be
highlighted over the scales of interest through the interrogation of the transform [35].
Consequently, the QRS complex can be distinguished from high P and T waves, noise,
baseline drift, and other artifacts in different scales. The relation between the
characteristic points of ECG signals and those of modulus maximum pairs of its wavelet
transforms are almost straightforward to establish [39].
22
Over the past few years, wavelets have been used in the analysis of physiological
signals, such as ECG, Electroencephalography (EEG), Electromyography (EMG), blood
pressure, and respiration signals [35]. Researchers have been exploring the applicability
of wavelets for capturing complex nonlinear dynamics of ECG [40-42]. However, not
much attention was given for the detection of certain complex patterns inside the ECG
signals. Conceivably, much of the information necessary for early diagnosis of various
ailments are buried in these complex patterns. In this dissertation research, an optimal
matched wavelet will be designed for ECG signals to find occurrences of certain complex
recurring patterns.
2.5 Databases
The research undertaken will use cardiovascular datasets from the PhysioNet for
the implementation and experiment studies. PhysioNet is a public service research
resource for complex physiologic signals [43].
The VCG data analyzed in myocardial infarction (MI) study (368 myocardial
infarction and 80 healthy control recordings) is from PTB database available from the
PhysioNet. The recordings were collected at the Department of Cardiology of University
Clinic Benjamin Franklin in Berlin, Germany. Each of these recordings contains 15
simultaneously recorded signals: the conventional 12-lead ECGs and the 3 Frank (XYZ)
VCG signals. Each of these is digitized at 1000HZ, with 16 bit resolution over a range of
±16.384 mV. The 80 healthy control recordings are from 54 healthy volunteers, and 368
MI recordings are from 148 patients. The recordings are typically about two minutes in
length, with a small number of shorter records (none less than 30 seconds).
23
The ECG data used in the atrial fibrillation (AF) study is from the 2004 PhysioNet
challenge “Spontaneous Termination of Atrial Fibrillation.” In this challenge,
classification needs to be made among the following three categories of AF patients test
signals:
1) Group N: Non-terminating AF (defined as AF that was not observed to have
terminated for the duration of the long-term recording, at least an hour following
the segment).
2) Group S: Soon to be terminating (AF that terminates one minute after the end of
the record).
3) Group T: Terminating immediately (AF terminating within one second after the
end of the record).
In all, 80 recordings of AF from 60 different subjects were made available in the
database [43]. Each record is a one-minute segment of AF, containing two ECG signals,
each at 128 samples/sec. The data is divided into a learning set and two test sets. The
learning set contains 30 records in all, with 10 records in each of the three groups. The
learning set records were obtained from 20 different subjects (10 from group N and 10
from group S/T). Test set A contains 30 records from 30 subjects. About half of these
records belong to group N and the rest to group T. The goal is which records in test set A
belong to group T. Test set B contained 20 records, 2 from each of 10 subjects (none
represented in the learning set or in test set A). One record of each pair belongs to group
S and the other to group T. The goal of the second challenge event is to identify which
records belong to group T.
24
2.6 References
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of Bioelectric and Biomagnetic Fields. USA: Oxford University Press, July 18,
1995.
[2] D. Dubin, Rapid Interpretation of EKG's: An Interactive Course: Cover
Publishing Company, 2000.
[3] Marquette-KH, "ECG Electrical and Mechanical Events Diagram," Marquette
Electronics, 1996.
[4] G. E. Dower and H. B. Machado, "XYZ data interpreted by a 12-lead computer
program using the derived electrocardiogram," Journal of electrocardiology, vol.
12, pp. 249-261, 1979.
[5] G. E. Dower, H. B. Machado, and J. A. Osborne, "On deriving the
electrocardiogram from vectorcardiographic leads," vol. 3, p. 87, 1980.
[6] G. E. Dower, A. Yakush, S. B. Nazzal, R. V. Jutzy, and C. E. Ruiz, "Deriving the
12-lead electrocardiogram from four (EASI) electrodes," Journal of
electrocardiology, vol. 21, pp. S182-S187, 1988.
[7] A. Benchimol, F. Reich, and K. B. Desser, "Comparison of the electrocardiogram
and vectorcardiogram for the diagnosis of left atrial enlargement," Journal of
electrocardiology, vol. 9, pp. 215-218, 1976.
[8] I. Black, "The electrocardiogram and vectorcardiogram in congenital heart
disease," The Journal of Pediatrics, vol. 69, pp. 846-716, 1966.
[9] B. W. McCall, A. G. Wallace, and E. H. Estes, "Characteristics of the normal
vectorcardiogram recorded with the Frank lead system," The American Journal of
Cardiology, vol. 10, pp. 514-524, 1962.
25
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Surface ECGs Recorded During Atrial Fibrillation," Computers in Cardiology,
vol. 29, pp. 21-24, 2002.
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Journal of electrocardiology, vol. 33, pp. 176-S74, 2000.
[12] D. M. Schreck, J. D. Frank, E. R. Malinowski, L. E. Thielen, A. F. Bhatti, A. R.
Rivera, and V. J. Tricarico, "Mathematical modeling of the electrocardiogram and
vectorcardiogram using factor analysis," Annals of Emergency Medicine, vol. 23,
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[13] E. Simonson, N. Tuna, N. Okamoto, and H. Toshima, "Diagnostic accuracy of the
vectorcardiogram and electrocardiogram : A cooperative study," The American
Journal of Cardiology, vol. 17, pp. 829-878, 1966.
[14] K. K. Talwar, S. Radhakrishnan, V. Hariharan, and M. L. Bhatia, "Spatial
vectorcardiogram in acute inferior wall myocardial infarction: its utility in
identification of patients prone to complete heart block," International Journal of
Cardiology, vol. 24, pp. 289-292, 1989.
[15] A. van Oosterom, Z. Ihara, V. Jacquemet, and R. Hoekema, "Vectorcardiographic
lead systems for the characterization of atrial fibrillation," Journal of
electrocardiology, 2006.
[16] J. L. Willems, E. Lesaffre, and J. Pardaens, "Comparison of the classification
ability of the electrocardiogram and vectorcardiogram," The American Journal of
Cardiology, vol. 59, pp. 119-124, 1987.
[17] F. N. Wilson and F. D. Johnston, "The vectorcardiogram," American Heart
Journal, vol. 16, pp. 14-28, 1938.
[18] A. C. Witham, "Quantitation of the vectorcardiogram," American Heart Journal,
vol. 72, pp. 284-286, 1966.
26
[19] S. T. S. Bukkapatnam, S. Kumara, and A. Lakhtakia, "Real-time chatter control in
machining using chaos theory," CIRP Journal of Manufacturing Systems, vol. 29,
pp. 321-326, 1999.
[20] S. T. S. Bukkapatnam, S. R. T. Kumara, and A. Lakhtakia, "Dependence of
computed trajectories on the step size in a nonlinear dynamic system: An
investigation into cutting tool dynamics," IIE Transactions, vol. 27, pp. 519-529,
1995.
[21] S. T. S. Bukkapatnam, S. R. T. Kumara, and A. Lakhtakia, "Local
Eigenfunctions-Based Suboptimal Wavelet Packet Representation of
Contaminated Chaotic Signals," IMA Journal of Applied Mathematics, vol. 63,
pp. 149-160, 1999.
[22] S. T. S. Bukkapatnam, S. R. T. Kumara, A. Lakhtakia, and P. Srinivasan,
"Neighborhood Method and its Coupling with the Wavelet Method for Nonlinear
Signal Separation of Contaminated Chaotic Time-Series Data," Signal
Processing, vol. 82, pp. 1351-1374, 2002.
[23] S. T. S. Bukkapatnam, A. Lakhtakia, and S. R. T. Kumara, "Analysis of sensor
signals shows turning on a lathe exhibits low-dimensional chaos," Physical
Review E, vol. 52, pp. 2375-2387, 1995.
[24] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis. Cambridge:
Cambridge University Press, 1997.
[25] F. Takens, Detecting strange attractors in turbulence, lecture notes in
mathematics vol. 898: Springer, Berlin, 1981.
[26] H. Yang, S. T. Bukkapatnam, and R. Komanduri, "Nonlinear adaptive wavelet
analysis of electrocardiogram signals," Physical Review E (Statistical, Nonlinear,
and Soft Matter Physics), vol. 76, p. 026214, 2007.
27
[27] M. B. Kennel, R. Brown, and H. D. I. Abarbanel, "Determining embedding
dimension for phase-space reconstruction using a geometrical construction,"
Physical Review A, vol. 45, p. 3403, 1992.
[28] S. T. S. Bukkapatnam, "Recurrence-based piecewise eigen representation of
contaminated chaotic signals," Phys Rev E (under review), 2007.
[29] N. Marwan, N. Wessel, U. Meyerfeldt, A. Schirdewan, and J. Kurths,
"Recurrence plot based measures of complexity and their application to heart-
rate-variability data," Physical Review E, vol. 66, Aug 2002.
[30] N. Thomasson, T. J. Hoeppner, C. L. W. Jr., and J. P. Zbilut, "Recurrence
quantification in epileptic EEGs," Physics Letters A, vol. 279, pp. 94-101, 2001.
[31] J. P. Zbilut, A. Giuliani, and C. L. Webber, "Recurrence quantification analysis as
an empirical test to distinguish relatively short deterministic versus random
number series," Physics Letters A, vol. 267, pp. 174-178, 2000.
[32] Y. Zhu and N. V. Thakor, "Adaptive recurrent filter for ectopic beat and
arrhythmia detection," IEEE Engineering in Medicine and Biology Society 10th
Annual International Conference, 1988.
[33] L. Matassini, H. Kantz, J. Holyst, and R. Hegger, "Optimizing of recurrence plots
for noise reduction," Physical Review E, vol. 65, p. 021102, 2002.
[34] N. Marwan, M. Carmen Romano, M. Thiel, and J. Kurths, "Recurrence plots for
the analysis of complex systems," Physics Reports, vol. 438, pp. 237-329, 2007.
[35] P. S. Addison, "Wavelet Transforms and the ECG: A Review," Physiology
Measurement, vol. 26, pp. 155-199, 2005.
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Cambridge University Press, 2000.
28
[37] G. Strang and T. Nguyen, Wavelets and Filter Banks: Wellesley-Cambridge
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[38] C. Valens, "A Really Friendly Guide To Wavelets," 1999-2004.
[39] C. Li, C. Zheng, and C. Tai, "Detection of ECG Characteristic Points Using
Wavelet Transforms," IEEE Transactions on Biomedical Engineering, vol. 42, p.
21, 1995.
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Contemporary Physics, vol. 40, pp. 31 - 55, 1999.
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G. Moody, C. Peng, and H. Stanley, "PhysioBank, PhysioToolkit, and PhysioNet:
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Circulation 101, vol. 23, pp. e215-e220, 2000 (June 13).
29
CHAPTER III
RESEARCH METHODOLOGY
This chapter outlines the research methodology used in subsequent chapters. A
schematic of the methodology for developing cardiovascular monitoring and prognostic
schemes from the study of nonlinear stochastic dynamics underlying the ECG signals is
shown in Figure 3.1. Its five modules can be divided as follows:
� Signal representation
� Feature extraction and state classification
� Simulation and prognostic modeling
Figure 3.1 Schematic of research methodology
30
3.1 Signal representation
In the first part of research methodology, cardiovascular signals are presented in
alternative domains, such as time, frequency, time-frequency domains, or state space, so
that mathematical description of the salient patterns contained in the signal as well as the
procedures for feature extraction will be much simpler and efficient in the transformed
space.
3.1.1 Nonlinear adaptive wavelet representation of ECG signals
Wavelet representation can provide an effective time-frequency analysis for
nonstationary signals, such as the electrocardiogram (ECG) signals, which contain both
steady and transient parts. In recent years, wavelet representation has been emerging as a
powerful time-frequency tool for the analysis and measurement of ECG signals. The
ECG signals contain recurring, near-periodic patterns of P, QRS, T, and U waveforms,
each of which can have multiple manifestations. Identification and extraction of a
compact set of features from these patterns is critical for effective detection and diagnosis
of various ailments. This section will present an approach to extract a fiducial pattern of
ECG based on the consideration of the underlying nonlinear dynamics. The pattern, in a
nutshell, is a combination of eigenfunctions of the ensembles created from a Poincare
section of ECG dynamics. The adaptation of wavelet functions to the fiducial pattern thus
extracted yields two orders of magnitude (some 95%) more compact representation
(measured in terms of Shannon signal entropy). Such a compact representation can
facilitate in the extraction of features that are less sensitive to extraneous noise and other
variations. The adaptive wavelet can also lead to more efficient algorithms for beat
31
detection and QRS cancellation as well as for the extraction of multiple classical ECG
signal events, such as widths of QRS complexes and QT intervals.
3.1.2 Dynamic VCG representation
Vectorcardiogram (VCG) signals monitor the spatiotemporal cardiac electrical
activity along three pseudo-orthogonal planes, namely, frontal, transverse, and sagittal
planes of the body. However, the absence of temporal resolution in the conventional
static representation of VCG signals is a major impediment to a medical doctor in the
interpretation and clinical usage of VCG. This is especially so since cardiologists depend
heavily on the intervals and segment durations in the ECG signals, i.e. temporal
resolution. On a paper, only three orthogonal X,Y,Z lead vectors can be shown, and the
fourth dimension, namely, the temporal scale can not be shown as it constitutes the fourth
dimension. Alternately, a combination of any three vectors can be shown on a paper. In
this investigation, we present an approach that captures the critical spatiotemporal heart
dynamics by displaying VCG signals on a computer screen, instead of on a paper, the real
time motion of a cardiac vector in a 3-D space. Such a dynamic display of 3-D heart
activity can be realized even with one lead ECG signal (e.g., ambulatory ECG) through
an alternative lag reconstructed spatiotemporal EKG attractor representation from
nonlinear dynamics principles. The VCG representation can also be color coded for
spatiotemporal representation with additional heart dynamical properties of cardiac
vector movements, such as the curvature, velocity, and phase angle. The dynamic VCG
representation is shown to significantly enhance the interpretability of vectorcardiogram
and facilitate the identification of related spatiotemporal cardiovascular dynamics.
32
3.2 Feature extraction and classification
The second part of research methodology involves the extraction of invariant
dynamical quantifiers that describe the specific patterns in the signal and are sensitive to
the state variables to be estimated. Moreover, in the classification stage, the extracted
features are associated with an appropriate representation of unknown state variables. In
this part, atrial fibrillation and myocardial infarction will be studied.
3.2.1 Atrial fibrillation state classification
Atrial fibrillation (AF), in general, can sustain indefinitely since the ventricles
continue to perform the essential function of blood circulation, albeit inefficiently.
Nevertheless, the risks of sustained AF can be serious, and may include strokes and
myocardial infarction (MI) caused by the formation of blood clots within stagnant
volumes in the atria. AF may be subdivided into three different forms, namely,
paroxysmal AF (self terminating), persistent AF, and permanent AF. This section
presents the results of signal analysis and a classification technique, namely,
Classification and Regression Tree (CART), for detecting spontaneous termination or
sustenance of AF.
3.2.2 Myocardial infarction identification using recurrence quantification analysis of
VCG signals
Myocardial infarction (MI), also known as heart attack, is one of the leading causes
of death in the world. It is resulted from the occlusion of the coronary artery and
insufficient blood supply to myocardium. Locations of myocardial infarction can be in
inferior, septal, anteroseptal, anterior, apical, and other portions of the heart. The necrosis
33
or death of myocardial cells in a certain location of the heart will lead to changes in the
underlying cardiovascular dynamics.
Heart monitoring signals, such as the conventional 12-lead electrocardiogram (ECG)
and 3-lead (Frank XYZ) vectorcardiogram (VCG) provide valuable information on the
underlying dynamic activities from different perspectives of the heart. Dower
demonstrated that a 12-lead ECG signals can be linearly transformed to 3-lead
vectorcardiogram, which is commonly known as Dower transformation. It is, therefore,
imperative to analyze the VCG signals and explore its relationships with cardiovascular
disorders just in the same way as 12-lead ECG is used in the clinical practice. Techniques
from nonlinear dynamical system analysis, such as recurrence quantification analysis
(RQA) provide effective means to establish these relationships. This section presents the
application of RQA for detecting myocardial infarction (MI) from VCG signals.
Significant differences were observed in the laminarity, determinism, and local
divergence of VCG trajectories between Healthy Control (HC) subjects and MI patients.
MI classification accuracies based on the extracted nonlinear dynamical invariants from
the VCG recurrence plots were found to be as high as 97% for neural network and linear
regression classification model.
3.2.3 Vectorcardiographic octant features for the diagnostics of heterogeneous
myocardial infarction
Vectorcardiogram (VCG), three orthogonal cardiac components, was reported to be a
promising candidate for the diagnostics of MI. But few, if any, of previous VCG
diagnostic approaches could achieve > 90% accuracy in both specificity and sensitivity.
34
This investigation assesses new vectorcardiographic octant features towards the effective
detection of heterogeneous myocardial infarction.
VCG octant features and conventional electrocardiogram (ECG) interval features
were extracted from 448 VCG recordings in Physionet PTB database, which includes 368
MIs and 80 HCs. Statistical feature analysis shows that cardiac vector length distributions
in octant 0 (x-negative, y-negative, z-negative) and octant 2 (x-negative, y-positive, z-
negative), T vector length and octant position are the most prominent features to
distinguish MIs from HCs. It is significant at 69% level (p-value=9.63E-29) for the
Oct0MaxN (the maximum vector length in octant 0) distribution differences between MI
(0.38±0.16) and HC (0.54±0.04). The majority (74/80) of HCs’ T vector positions
(T_pos) is also found to cluster in octant 6 (x-positive, y-positive, z-negative), while MIs
spread over all the eight octants. Additionally, Oct2AvgN (the average vector length in
octant 2) and Oct0VarN (the variations of all vector lengthes in octant 0) show >60%
statistical distribution differences. With only four octant features (Oct0MaxN, T_pos,
Oct2AvgN and Oct0VarN), a simple Classification and Regression Tree (CART) model
can yield classification accuracy with 96.74% sensitivity and 96.25% specificity. The
stochastic experiments with different percentage of training data size also reveal high
sensitivity (mean: 96%) and specificity (mean: 82%) for heterogeneous MI and HC
classification.
Octant 0, octant 2, and T vector information in vectorcardiogram are found to be very
important for the diagnostics of heterogeneous MIs. Random classification experiments
demonstrate the generality and effectiveness of CART model and vectorcardiographic
35
octant features. This study is definitely indicative of potential clinical applications of MI
diagnostic model from the VCG octant features.
3.3 Simulation and prognostic modeling
The third section will focus on the cardiovascular disorder simulation and prognostic
modeling. The mathematical prediction model can be used for prognosis of the patient’s
health status with the online monitoring ECG signals, and it updates the prediction results
in real time and provides the instantaneous alarm for the dangers. Such prognostic results
help determine whether it makes more sense to attempt certain treatments or to withhold
them, and thus play an important role in medical decision support. In this section, a local
recurrence prediction model will be developed to accommodate the nonlinear and
nonstationary environments in the complex physiological process. Next, Radial Basis
Function (RBF) simulation model for the VCG signals is shown to simulate the patient’s
cardiovascular electrical activities from a data-driven’s perspective.
3.3.1 Local recurrence prediction in nonstationary chaotic systems
A local recurrence modeling approach that considers the underlying nonlinear
stochastic dynamics and nonstationarities is designed for predicting the state of complex
physical systems. This approach uses heretofore unexplored premise of treating
nonstationarity as to cause finite time detours of system dynamics from an attractor’s
vicinity. The local recurrence property, exhibited over a certain sets of time and state
space segments, is used to partition system trajectory into multiple near-stationary
segments. Consequently, local eigen representations of each segment can capture both
dynamics and nonstationarities. Extensive studies using simulated and real world datasets
36
reveal significant prediction accuracy improvements over other conventional alternative
methods.
3.3.2 Radial basis function simulation modeling of VCG signals
The vectorcardiogram (VCG) represents human’s heart potential variations and
regulations in the vector form, and it has been shown to be very important for some
disease assessments. This section presents a general data-driven method for the modeling
and simulation of human VCG and standard 12-lead electrocardiogram (EKG) signals.
The heartbeat variations and VCG X, Y, Z ensemble components are extracted by the
Poincare sectioning of the VCG trajectory. The Heart Rate Variability (HRV) is modeled
as a Weibull distribution, and Radial Basis Function (RBF) network model is trained with
experiment data to simulate the VCG trajectory and thus duplicate the heart electric
activity. The parameterized RBF network model is used for the generation of simulated
VCG signals and they are transformed to get the standard 12-lead EKG signals by the
generalized Dower transformation matrix or the estimated matrix from experiment data
for a specific patient. This investigated RBF network model is shown to closely capture
the same heart activities as the actual data in time domain, frequency domain. Such a
well-trained data-driven model can facilitate the study of VCG, feature extraction from
VCG and 12-lead EKG signals, and further prognosis of diseases.
PART I
SIGNAL REPRESENTATION
37
CHAPTER IV
CUSTOMIZED WAVELET REPRESENTATION OF ECG SIGNALS
Patterns from the surface electrocardiogram (ECG) signals are widely used in the
diagnosis of various cardiac disorders [1-3]. Similar to the pattern recognition techniques
used successfully in speech, fingerprinting etc. [4, 5], ECG signal pattern matching may
also provide the opportunity to recognize the same person (biometric identification) and
diagnose heart diseases by means of the similarity comparison with the evaluated patterns
of a typical patients’ ECG in the collected databases. This chapter presents a new
technique for customizing the wavelet functions adapted to the ECG signal pattern
through the use of nonlinear dynamic principles. In specific, we show that using tools
such as Poincare section, one can achieve extremely compact signal representation (two
orders of magnitude reduction in signal entropy) that can be more sensitive to different
ECG signal patterns. The remainder of chapter 4 is organized as follows: Section 4.2
gives a brief background of the applications of wavelet transform in surface ECG
analysis; Section 4.3 is the phase space fiducial pattern extraction; Section 4.4 is the least
squares matching wavelet design; Section 4.5 presents the implementation of designing a
customized ECG wavelet function and its comparison with standard wavelet library, and
Section 4.6 presents the conclusions from the reported research and perspectives on
future investigations.
38
4.1 Background
There are several existing families of standard wavelets, such as Haar, Daubechies,
Coiflet and so on [6]. The choice of wavelet basis functions plays a significant role in
determining the compactness of the resulting wavelet representation. There is no
consistent answer to the question: Which is the best wavelet? Some wavelets are better
suited for detecting some particular problems, such as discontinuities and breakdown
(e.g., Haar wavelet to detect discontinuities), while others are superior for long term
evolution or compression (for e.g., a sufficiently regular wavelet with at least k (k>3)
vanishing moments will be better to compactly represent a smooth signal).
Fidu
cial
patt
ern
extr
actio
n Phase space reconstruction
Poincare sectioning
Ensemble generation & eigen analysis
Figure 4.1 Procedure for customized wavelet design from system nonlinear dynamics
Nonetheless, it is generally understood that closer the basis functions match the
signal patterns, the more compact the representation will be. As shown in Figure 4.1, this
ECG Analyzer
Wav
elet
cust
omiz
atio
n Wavelet function form selection
Objective function formulation
Imposition of admissibility & constraints
Least square fitting
Customized wavelet
ECG time series
Fiducial pattern
39
chapter presents a customized wavelet function design using least square fitting for the
fiducial signal pattern extracted from the nonlinear system dynamic characterization of
signal state space. In the first step, system phase space trajectory will be reconstructed
from the observed time series. False nearest neighbor test and mutual information test
provide the necessary parameters, namely, embedding dimension dE and time delay
[7]. After the reconstruction of phase space, those ensembles can be extracted from
Poincare sectioning of the trajectory [8]. The Karhunen-Loeve (KL) transformation of
those extracted signal ensembles provides the fiducial signal pattern for the least square
customized wavelet design stage. We use polynomial wavelet structures and a
constrained least square fit procedure to fit the structure to match as closely as possible,
subject to admissibility and regularity constraints. The details of the two major steps of
fiducial pattern extraction and wavelet fitting are presented in the following two
subsections. Finally, representation entropy is calculated to compare the performance of
various wavelets. Signal entropy, h is a measure of parsimony of representation [9, 10].
Here, the normalized entropy with ECG signal energy is used as follows:
,
( , ) log ( , )s
h p s p s d ds�
� � � ��� (Eq 4-1)
where
2
2
( , )( , )
( )x s
p sx t
� ��
� , (Eq 4-2)
( , )x s� �� is a continuous wavelet transform coefficient at scale s, and translation �, and
is signal energy. The smaller the value of signal entropy, the greater is the
parsimony of the representation [11, 12]. Intuitively, “high entropy” means the
d�
2( )x t
40
representation coefficients is from a uniform (boring) distribution, i.e., the
histogram of distribution of values of coefficients would be flat, and “low entropy”
means that coefficients is from varied distribution (consisting of multiple
distinct peaks and large valleys) The varied distribution of representation provides the
accurate detection of signal pattern.
4.2 Phase space fiducial pattern extraction
Natural systems generally show nonstationary and complex behaviors. The
Karhunen-Loeve (KL) transform provides an optimal representation for second order
stochastic processes. The KL representation of a stochastic process x(t) is given by
( ) ( )j jj
x t b t!� . (Eq 4-3)
Here, jb ’s are KL representation coefficients, and the basis function ( )j t! are the linearly
independent solutions of
( , ) ( ) ( )j j jK t d t� ! � � " !�R , (Eq 4-4)
where K(t,�) is the autocovariance function calculated from a set of signal ensembles. It is
evident that are eigenvalues, and are eigenfunctions of K(t,�), and are therefore
orthogonal. The order of eigenvalues, highest to lowest, indicates the components in
order of significance. The KL basis can be approximated using a set of ensembles of the
process x(t) such that the mean square error (MSE) between a given set of ensembles and
their projection to the subspaces spanned by each basis function of the KL representation
is minimized.
( , )x s� ��
( , )x s� ��
j" ( )j t!
41
However, in the absence of multiple realizations of x(t), underlying nonlinear
dynamics should be taken into account in order to extract ensembles needed to develop
optimal representations (i.e., basis set) of signals from these systems. The nonlinear and
stochastic dynamic characterization of a system is usually helpful in providing
information on the dimensionality and the functional form of models that can capture the
observed behaviors [7].
Figure 4.2 An illustration of trajectories of an attractor intersecting a (planar) Poincare section
The time evolution of the phase space trajectories emanated from the system
explains the underlying nonlinear dynamics. Usually, the measured observations of a
process can not include all possible state variables. Couplings among the system’s
components imply that each single component contains necessary information about the
dynamics of the larger system. The embedding theorem of Takens [13] guarantees that
the reconstructed trajectory portrays the dynamics in the higher dimensional state space.
It states that a diffeomorphism exists between the reconstructed phase space with the
state vector given by
2 ( 1)( ) [ ( ), ( ), ( ),..., ( )]En n n d n d n d dx t x t x t x t x t� � � � (Eq 4-5)
Poincare section �
Trajectory
42
and original phase space, if , where dE is the embedding dimension, is the
time delay, and D is the dimension of the compact manifold containing the attractor. This
implies that the dimension and entropy spectra of the reconstructed attractor are the same
as those of the original one.
Poincare section is a dE-1 dimensional hyperplane intersecting with the phase space
trajectories (see Figure 4.2). The recurrence property of a chaotic attractor A shows that
for every and almost every A such that , in effect, the
trajectories with an attractor remain bounded. Those points , i =1, 2, … at which the
trajectory intersects the Poincare section follow a return map. For strictly periodic
trajectory, the points , i =1, 2, …will overlap (i.e., #�0) such that the duration between
to along the trajectory constitutes the period. For chaotic systems no two Pi’s
overlap. For near-periodic signals, such as ECG, each strand emanating from a Poincare
section intersection point Pi and lasting approximately till the next intersection Pi+1 along
the trajectory may be treated as a realization of a stochastic process from an invariant
probability space [12]. Due to heart rate variability, some ensembles move faster, i.e., the
two successive intersections occur over shorter intervals, compared to the others [14, 15].
In this investigation the length (time duration) of the ensembles is taken as the time
interval between the intersections of the fastest ensemble. Moreover, those collected
near-periodic ensembles provide an effective way for the Karhunen-Loeve (KL)
representation of this signal. In this ECG investigation, the largest eigenvalue is
4.3026 which implies that 98.34% of the total variation occurs along the leading (the first
one out of 41) eigenfunctions. Intuitively, each eigenfunction � ��! captures the shape of
2 1Ed D$ d�
0# � (0)x % , 0t& (0) ( )x x t #� �
iP
iP
iP 1iP
max"
43
a specific mode of variation (roughly, a degree of freedom) of x(�). The fiducial pattern
( )t' of a signal emerging from a process with d-degrees of freedom (or topological
dimension D, where d = �D�) is obtained as the optimal projection of the ensembles onto
the space spanned by � � � � � ���� d!!! �,, 21 . For computational convenience (during
matching wavelet design), the support of ( )t' is rescaled so that [0,1]t% .
4.3 Least square matching wavelet design
The continuous wavelet transform of the signal x(t) using the analysis wavelet
is:
(Eq 4-6)
where is the dual of [16]. If x(t) is similar to the wavelet basis functions, then the
coefficients will likely be large only for a few basis functions. Thus, the customized
wavelet function adapted to the ECG pattern may achieve the best performance. We
design the wavelet function using a least squares approach [6, 17, 18]. A wavelet basis
function is approximated as a polynomial regression of degree M:
(Eq 4-7)
Let us assume that N-sample time-series of fiducial ECG pattern be available
such that:
(Eq 4-8)
x��
( )t�
� �1( , )xt
ts x t dtss
� �� � �� �� � �� �� �
�� ( )t�
x��
( )t�
20 1 2
02
0 1 2
( ) ( )
( ) [1 ][ ]
MM M T
M mm
M
M
t a a t a t a t a t U t
U t t t ta a a a
� (
(
��
��
( )t'
{ ( ), 0 1}N n nZ t t' ) )
44
Similarly, we can reduce operator U(t) to a Vandermonde matrix A, whose elements are
powers of t, will be as following:
(Eq 4-9)
It is important that the function which can not only best fit the fiducial signal pattern
but also satisfy the wavelet admissibility and regularity conditions (Percival and
Walden11). The admissibility requirements for any valid real or complex-value
continuous-time function to be a wavelet basis function can be summarized as:
reconstruction, zero mean, finite energy and regularity constraints. The first three defines
the wave, and the last condition determines the rate of decay or the let. A function
satisfied with all the four conditions can be a valid wavelet for CWT [18].
Since the fiducial pattern is a finite length signal, the finite energy requirement
is automatically met. The zero mean condition implies that the Fourier
transform of vanishes at the zero frequency , where stands for
the Fourier transform of . So, it will also make sure that is finite
for the success of inverse continuous wavelet transform:
(Eq 4-10)
21 1 1
22 2 2
1
2
11
0 , 1
1
M
M
N
MN N N
t t tt t t
A t t
t t t
� �� �� � ) )� �� �� � �
��
�� � �
�
( )t�
( )t'
( )t�
( )t'
( ) 0t dt��
��
�
( )t�2
0( ) 0F�
��
( )F� �
( )t�2
( )FC d��
��
�
�
��
�
2
2
( )1 1( ) ( , ) ( )xs
Ftx t s d ds C dC s s
����
�
��� � � ��
�
��
� � � � �
45
Summarizing, the least square Mth degree polynomial fitting with regularity K for the
wavelet design will add zero mean and regularity conditions, and the deduction of revised
matrix and is as following.
(1) Zero mean
(Eq 4-11)
(2) Regularity
The moment of the function is defined as . If the function s first
i moments are zero , the number of vanishing moment of
the function is k+1.
(Eq 4-12)
New Vandermonde matrix will contain powers elements of t and the wavelet
constraints.
(Eq 4-13)
and (Eq 4-14)
A� NZ
12 11 100
0
( ) 02 1
MMa t a tt dt a tM
� � �
� � � �� �
thi ( )t� ( )it t dt� �
���
( ) 0 for 0it t dt j k� �
�� ) )�
( )t�
11 2 11 0 10
0
( ) 0 11 2 1
i i i Mi Ma t a t a tt t dt i k
i i i M�
� � ) )� � � �
� �
A�
21 1 1
2
1
1 ...
1 ...0 , 11 1/ 2 1/ 3 ... 1/( 1)
1/ 2 1/ 3 1/ 4 ... 1/( 2)
1/( 1) 1/( 2) 1/( 3) ... 1/( 1)
M
MN N N
N
t t t
t t tA t tM
M
k k k k M
� �� �� �� �� �
) ) � �� � � �� �� � �
� � �
� �
� � �
0 1 2[ ]Ma a a a( � 1[ ( ), , ( ),0,0, ,0]TN NZ t t' ' � �
46
The N-sample estimate of the coefficients vector of the matching polynomial wavelet
function can be determined to minimize the objective function
(Eq 4-15)
(Eq 4-16)
and the estimate can be obtained using the pseudoinverse:
(Eq 4-17)
4.4 Implementation and results
As discussed in the research methodology section, the coefficients of a compact
wavelet representation need to be more sensitive to variation in the underlying processes
(physiological) and less sensitive to noise variation. Thus, feature sets extracted from a
compact representation tend to be lighter (i.e. fewer and more sensitive) and more
effective in estimating various anomalies (here, different AF states). For implementation
and validation of the present approach, we utilized the ECG data from the 2004
PhysioNet challenge named Spontaneous Termination of Atrial Fibrillation (AF), posted
on the PhysioNet website [19, 20]. Atrial Fibrillation (AF) is one of the serious cardiac
disorders that affect millions of human beings, and its early detection can have a
significant bearing on the quality of healthcare. In this contest, classification needs to be
made among the following three categories of AF patients test signals:
1) Group N: Non-terminating AF (defined as AF that was not observed to have
terminated for the duration of the long-term recording, at least an hour following
the segment).
ˆN(
( , )N NV Z(
2
1
1( , ) ( )N
TN N N
tV Z Z A
N( (
�� �
ˆ arg min{ ( , )}N N NV Z(
( (
1( )T T TNZ A A A( � � � �
47
2) Group S: Soon to be terminating (AF that terminates one minute after the end of
the record).
3) Group T: Terminating immediately (AF terminating within one second after the
end of the record).
In all, 80 recordings of AF from 60 different subjects were made available in the
database. Each record is a one-minute segment of AF, containing two channel ECG
signals (Lead I and II), each at 128 samples/sec. Figure 4.3 contains ECG signal trace
taken from a subject with AF for the duration of approximately 10 heartbeats. The trace
shows a QRS complex with a significant R-peak followed by a T-wave. The signal is
superposed with higher frequency (>6Hz) atrial fibrillation F-waves in lieu of P-waves.
Figure 4.3 Time domain plot of a representative ECG signal trace (a01)
After the false nearest neighbor and mutual information test [7, 21], the optimal
embedding dimension and time delay were determined for reconstruction of the state
space from the ECG traces using time delay coordinates in dE embedding dimensions as
(see Figure 4.4). The reconstructed state
200 400 600 800
0
0.5
1
1.5
Time Index
EKG
Pot
entia
l (m
V)
2 ( 1)( ) [ ( ), ( ), ( ),..., ( )]En n n d n d n d dx t x t x t x t x t� � � �
48
space shows a large loop of QRS complex extending top-down and left to right, with the
R-peaks occurring at the top and right extrema. A T-loop as well as an irregular ball
formed by the F-waves are observed near the origin (bottom left corner). The Poincare
sections �1 and �2 of the state portrait, as shown in Figure 4.4 can be used to gather
alternative sets of pattern ensembles.
Figure 4.4 State space portrait reconstructed from time delay coordinates
The choice of the Poincare section � affects the shape of the pattern as well as the
performances (entropy) of the representation. For e.g., one needs to choose the Poincare
section such that the flow lines are directed about one or two flat planes, in which case �
takes the form of a simple planar object. Also, it is desirable that the local Lyapunov
exponent max* of the state is close to or below zero about �. This will help in making sure
the resulting wavelet easily satisfies the constraints imposed by Eqs. (4-11) and (4-12). In
this context, wavelet customized through Poincare section �1 can yield better
-0.5 0 0.5 1 1.5
-0.5
0
0.5
1
1.5
EKG potential (mV)
EKG
pot
entia
l (m
V)
Poincare sections
49
performances than that through Poincare section �2 as shown in Figure 4.4. Figure 4.5
shows the ensembles gathered from Poincare section �1. The fiducial ECG signal pattern
used for the wavelet design stage can be extracted from dominant eigenfunctions
estimated using the KL representation. Figure 4.6 shows the least square matching
wavelet design result. The resulting wavelet �(t) holds significant similarities to the
fiducial pattern '(t) and capture a majority of the variations among the ensembles.
Figure 4.5 ECG pattern ensembles extracted from Poincare section
Figure 4.6 Matching wavelet extracted from fiducial signal pattern
0 50 100 150 200
0
0.5
1
1.5
2
Time Index
EKG
pot
entia
l (m
V)
0 50 100 150
0
0.5
1
1.5
Time Index
EKG
Pot
entia
l (m
V)
Signal PatternMatched wavelet
R
S
TP/F
Q
Fiducial pattern of the signal
PCA
50
Figure 4.7 shows the variation of entropy (Eq. (4-1)) with the polynomial degree of
the wavelet function �(t) that matches the fiducial pattern. The minimal entropy for
matching wavelet (shown in Figure 4.7) is found to have a polynomial degree of 13.
Figure 4.7 Matching wavelet representation entropy variation with various polynomial
degrees
As summarized in Figure 4.8, the customized wavelets an ECG from a non-
terminating AF case, obtained through Poincare sections �1 and �2 of the reconstructed
state space, denoted a01pat_pi1 and a01pat_pi2, respectively, and customized wavelets
from normal ECG pattern (nmpat_pi1 and nmpat_pi2)), shown in the last four bars on the
right side, are about 95% (approximately two orders of magnitude) more compact than all
standard available wavelet bases investigated (shown on the left side), namely, db2
(Daubechies-2), sym2 (Symlet-2), haar (Haar), mexh (Mexican hat), meyr (Meyer),
dmeyr (Discrete Meyer), morl (Morlet), db10 (Daubechies-10), sym10 (Symlet-10). Also,
the wavelets customized for a normal ECG signal seems to yield more compact
0 10 20 30 40 50 600.246
0.248
0.25
0.252
0.254
0.256
Polynomial Degrees
Ent
ropy
Minimum in 13thpolynomial degree
13
51
representations for a signal from a non-terminating AF case (i.e., a01 signal in the
PhysioNet [20].
Figure 4.8 Entropy comparison between standard wavelets and matching wavelets
The two orders of magnitude increase in the compactness of ECG signal
representation with customized wavelet (measured in terms of entropy reduction) is
further evidenced from the examination of the distribution of entropies for eleven signals
(a01, a02, a03, b01, b02, n01, n02, s01, s02, t01, t02) in the PhysioNet database for the
AF challenge with the eleven alternative wavelet bases including the standard and the
customized wavelets (see Figure 4.9). Interestingly, the wavelets customized for an ECG
signal from a normal case yield about five times lower entropy compared to that from a
non-terminating AF case. Also, it may be noted that, although both customization yield
ignorantly low entropy compared to standard wavelet basis, the entropy of wavelet
adapted from Poincare section �2 is twice as large as that from Poincare section �1. It is
0
5
10
15
20
db2
sym
2
haar
mex
h
mey
er
dmey
er
mor
let
db10
sym
10
a01p
at_p
i1
nmpa
t_pi
1
a01p
at_p
i2
nmpa
t_pi
2
Entr
opy
0.248
0.081
0.080.439
52
evidently due to the fact that the customized wavelet from �1 naturally closes to
satisfying the wavelet constraint Eqs. (4-11) and (4-12).
Figure 4.9 Entropy distribution for different wavelet representations, taken over all ECG signals in the database considered
Reasons for the compactness of the representations are further evident from an
examination of the histograms of wavelet coefficients of the representations (see Figure
4.10). The figure shows the distribution of wavelet coefficients within various bins of the
histogram. The coefficients from the customized wavelet representation (see Figure 4.10
(a)) are concentrated around zero with few large coefficients. Such a low entropy
distribution, with few large coefficients and several near-zero coefficients can lead to
clearer demonstration and identification of various salient events in ECG signals. In
contrast, coefficients of a representation from a standard library wavelet (see Figure 4.10
(b)) are spread uniformly, providing no clear distinction between significant and non-
0
5
10
15
20
25En
trop
y
db2
sym
2
haar
mex
h
mey
er
dmey
er
mor
let
db10
sym
10
a01p
at_p
i1
nmpa
t_pi
1
a01p
at_p
i2
nmpa
t_pi
2
0.224±0.073
0.074±0.0240.368±0.12
0.073±0.025
53
significant coefficients. Therefore, we have used the customized wavelets for QRST
cancellation and feature extraction.
Figure 4.10 Histogram of matching wavelet transform coefficients: (a) customized wavelet, (b) Morlet wavelet
4.5 Summary
The choice of various basis functions is known to determine the compactness of a
wavelet representation. In general, the closer the basis function captures the signal
characteristics, the more compact is the representation, and more likely are the features
sensitive to relevant ECG states and insensitive to variations in extraneous noise. In this
chapter, we have customized the basis functions of a continuous wavelet representation
by choosing polynomial wavelet basis functions that match the characteristics of a
fiducial 1-beat long ECG signal pattern extracted from the Poincare sectioning of ECG
state space. The customized representations were found to be roughly two orders of
magnitude more compact (measured in term of signal entropy) than the wavelet basis
functions available in the standard wavelet library. The unraveling of scale-time
distribution of signal content in wavelet representation can facilitate the identification of
Wavelet Coefficients, ( , )x s� �� Wavelet Coefficients, ( , )x s� ��
Freq
uenc
y
Freq
uenc
y
54
various ECG events including the onsets, peaks, and offsets of various ECG waveforms
for different beats. Further, it provides a means for QRST subtraction to suppress ECG
signal components that emerge from ventricular sources.
4.6 References
[1] Z.D.Yum, J.Q.Xu, and G.P.Li, "Recognition of cardiac patterns based on wavelet
analysis," in Proceedings of tne 2003 IEEE International Symposium on Intellgent
Control, Houston, Texas, October 5-8. 2003, pp. 642-645.
[2] Z. Huang, "Genetic Algorithm Optimized Feature Extraction and Selection for
ECG Pattern Classification," in Department of Electrical and Computer
Engineering. vol. M.S. Thesis: Michigan State University, 2002.
[3] R. Bousseljot and D. Kreiseler, "ECG Signal Analysis by Pattern Comparison,"
Computers in Cardiology, vol. 25, pp. 349-352, 1998.
[4] A. K. Jain, R. P. W. Duin, and J. Mao, "Statistical Pattern Recognition: A
Review," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.
22, pp. 4-37, January 2000.
[5] T. Irino and H. Kawahara, "Signal Reconstruction from Modified Wavelet
Transform - An application to Auditory Signal Processing," IEEE Transactions
on Signal Processing, vol. 41, pp. 3549-3554, 1993.
[6] D. B. Percival and A. T. Walden, Wavelet Methods for Time Series Analysis:
Cambridge University Press, 2000.
[7] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis. Cambridge:
Cambridge University Press, 1997.
[8] S. T. S. Bukkapatnam, "Recurrence-based piecewise eigen representation of
contaminated chaotic signals," Physical Review E (under review), 2007.
[9] M. V. Wickerhauser, "Lectures on Wavelet Packet Algorithms," Nov. 18, 1991.
55
[10] R. R. Coifman and M. V. Wickerhauser, "Entropy-Based Algorithms for Best
Basis Selection," IEEE Transactions on Information Theory, vol. 38, pp. 713--
718, 1992.
[11] S. T. S. Bukkapatnam, "Compact nonlinear signal representation in machine tool
operations," in ASME Design Engineering and Technology Conference, DETC-
VIB 8068, Las Vegas, NV, September 12-15, 1999.
[12] S. Bukkapatnam, S. Kumara, and A. Lakhtakia, "Local eigenfunctions-based
suboptimal wavelet packet representation of contaminated chaotic signals," IMA
Journal of Applied Mathematics, vol. 63, pp. 149-160, 1999.
[13] F. Takens, Detecting strange attractors in turbulence, lecture notes in
mathematics vol. 898: Springer, Berlin, 1981.
[14] F. Marciano, M. L. Migaux, D. Acanfora, G. Furgi, and F. Rengo, "Quantification
of Poincare maps for the evaluation of heart rate variability," 1994, pp. 577-580.
[15] G. D'Addio, G. D. Pinna, R. Maestri, G. Corbi, N. Ferrara, and F. Rengo,
"Quantitative Poincare plots analysis contains relevant information related to
heart rate variability dynamics of normal and pathological subjects," 2004, pp.
457-460.
[16] I. Daubechie and B. Han, Pairs of Dual Wavelet Frames from Any Two Refinable
Functions. New York: Springer, May 12, 2004.
[17] C. Valens, "A Really Friendly Guide to Wavelets," 1999-2004.
[18] G. Strang and T. Nguyen, Wavelets and Filter Banks: Wellesley-Cambridge
Press, 1996.
[19] G. Moody, "Spontaneous Termination of Atrial Fibrillation: A Challenge from
PhysioNet and Computers in Cardiology 2004," Computers in Cardiology, vol.
31, pp. 101-104, 2004.
56
[20] A. Goldberger, L. Amaral, L. Glass, J. Hausdorff, P. Ivanov, R. Mark, J. Mietus,
G. Moody, C. Peng, and H. Stanley, "PhysioBank, PhysioToolkit, and PhysioNet:
Components of a New Research Resource for Complex Physiologic Signals,"
Circulation 101, vol. 23, pp. e215-e220, 2000 (June 13).
[21] S. Bukkapatnam, A. Lakhtakia, and S. Kumara, "Analysis of sensor signals shows
turning on a lathe exhibits low-dimensional chaos," Physical Review -E., pp.
2375-2387, 1995.
57
CHAPTER V
DYNAMIC VCG REPRESENTATION
The conventional 12-lead electrocardiogram (ECG) and the 3-lead Frank X,Y,Z
vectorcardiogram (VCG) signals are recorded at the body surface to monitor the
underlying heart’s electrical activities and used to provide valuable information for the
medical diagnosis of the cardiovascular condition of a patient. The heart activities are
monitored from various perspectives by these signals. The 12-lead ECG is more
commonly used than the 3-lead VCG because medical doctors have been accustomed to
using them in clinical diagnosis for more than one hundred years. It has thus proven its
value, time tested, and considered as the Gold Standard. However, much of that
information is redundant and even in that, only a small fraction of the data is used in the
analysis by the physicians based on experience and expertise and oftentimes on the
memorization of ECG signals for different cardiological disorders. This is a difficult task
and the cardiologists are constantly looking for alternatives.
VCG monitors the cardiac electrical activity in three pseudo-orthogonal X,Y, Z
planes of the body, namely, frontal, transverse, and sagittal (see Figure 2.3) [1]. It is
traditionally projected to different planes (X-Y plane, X-Z plane and Y-Z plane) which
captures the time correlations, or plot as a static attractor in a 3D space that provides the
topological relationships. The absence of combined spatiotemporal information in the
58
VCG representations reported earlier, expects the interpreters to have not only the spatial
but also the temporal imaginations.
This chapter presents a dynamic VCG representation approach to capture the
spatiotemporal dynamics underlying the cardiac electrical activities. Instead of reading
signals from a paper which at most gives three dimensions, the monitor of the computer
is used to view all four dimensions, namely, the three X, Y, and Z components and one
time scale. The example VCGs used in this investigation are gathered from the PTB
database, which is available from the PhysioNet – a public service of the research source
for the complex physiological signals [2]. Each of the recordings in the database contains
15 simultaneously recorded signals, namely, the conventional 12-lead ECGs and the three
orthogonal Frank XYZ leads VCG signals digitized at 1 kHz, with a 16 bit resolution
over a range of ±16.384 mV.
The remainder of this chapter is organized as follows: Section 5.1 presents a brief
background on the linear transformation between 3-lead VCG and the 12-lead ECG;
Section 5.2 provides the research methodology used; Section 5.3 presents a discussion of
the pathological patterns in VCG signals, and Section 5.4 covers the conclusions based
on the research reported here.
5.1 Background
The electrocardiogram, designed by Augustus Waller in 1889 and further improved
by Williem Einthoven in 1901, has been used for more than 100 years for the clinical
diagnosis of cardiovascular disorders. Einthoven denoted the P, QRS, and T waves in the
EKG signal to the sequences of atrial and ventricular depolarization and repolarization
activities. Around 1904 Einthoven et al. [3] derived the famous “Einthoven triangle” to
59
measure three EKG signals and calculate the approximate direction of the cardiac vector.
In 1956, Ernest Frank designed the Frank lead system to measure the VCG in three
corrected orthogonal coordinates, which provides a clear picture of the cardiac vector.
With the rapid development of information technology (IT) and the availability of
computing hardware at reasonable cost, the representation and analysis of 3D VCG loops
are not constrained by computational resources and this has resulted in renewed interest
in VCG since 1990’s. Dower and his colleagues [4-6] conducted pioneering research
based on Frank’s tank torso model studies and introduced a linear transformation matrix
to convert 3-lead VCG signals into 12-lead ECG signals without a significant loss of
clinically useful information regarding heart dynamics. The inverse Dower
transformation matrix is used to derive the 3-lead VCG from the 12-lead ECG. Some
even consider Dower transformation matrix as generalized or universal transformation
matrix [4-7] although this has not been established to date and one finds different
transformation matrices for healthy subjects and patients with cardiological disorders [8].
The transformation studies show the statistical equivalence between the 12-lead
ECG and the 3-lead VCG. However, it is difficult for the human beings to visually
project a spatial VCG vector into any specified cardiac measurement angle determined by
the 12-lead measurement systems, which is the conventional way for the ECG
interpretation. In addition, the absence of temporal information in the static VCG
representation poses extra barriers for medical doctors to closely relate the 12-lead ECG
characteristics to the 3-lead Frank X, Y, Z VCG signals. But when it comes to the
computer automated analysis of heart monitoring system, the 12-lead ECG signals will
introduce the “curse of dimensionality” problem, which is the convergence of any
60
estimator to the true value of a smooth function defined on a space of high dimension is
very slow because of the exponential increase in volume associated with extra
dimensions. Thus, VCG is a better option for computer processing and analysis because it
overcomes not only the loss of information from the analysis of only one or two ECG
signals but also the dimensionality problems induced by the 12-lead ECG. The usage of
VCG for the medical diagnosis has been studied by many researchers [9-20] although
investigations exploring both spatial and temporal relationships of the VCG signals are
rarely, if any, found. It is hoped that present approach of spatiotemporal VCG
representation would greatly improve the exploration of spatiotemporal cardiovascular
dynamics and lead to automated computer analysis. This approach will be introduced in
the following sections.
5.2 Research methodology
The three VCG vector loops, namely, P, QRS, and T waves contain three
dimensional recurring, near-periodic patterns of heart dynamics. The dynamic VCG
representation can provide doctors with an easier way to understand, interpret, and use
vectorcardiograms. This approach includes presenting VCG signals as a real time motion
of cardiac vector in 3D space, and color coding of the magnitude of the cardiac vector
movement with the curvature, velocity, and phase angle etc. In addition, an alternative
lag reconstructed heart attractor representation from nonlinear dynamics principles is
provided, if there is only one dimensional heart monitoring signal available. The
proposed dynamic representation of the vector loops can be rotated freely on the monitor
to analyze the loop’s form in real time.
61
5.2.1 Spatiotemporal VCG attractor representation
In the Frank XYZ lead system, a vectorcardiogram is represented as three
orthogonal scalar measurements with respect to time as shown in Eq 5-1. The dynamic
VCG representation embeds the cardiac vector, composed of three scalar measurements
as x, y, z components, in real time. As shown in Figure 5.1, the representation is divided
into two parts. The top half plots the three scalar x, y, z components in real time and the
bottom half presents the movement of cardiac vector in 3D space simultaneously.
( )( )( )
x
y
z
v f tv g tv h t
+, -, .
(Eq 5-1)
62
P
atie
nt00
5\s0
101l
re: M
yoca
rdia
l inf
arct
ion,
ant
erio
r
-0.2
00.
20.
40
0.2
0.4
0.60
0.2
0.4
0.6
VxV
y
Vz0
500
1000
1500
2000
2500
3000
Vz Vy Vx
(ms)
-0.2
00.
20.
40
0.2
0.4
0.60
0.2
0.4
0.6
Vx
Vy
Vz0
500
1000
1500
2000
2500
3000
Vz Vy Vx
(ms)
-0.2
00.
20.
40
0.2
0.4
0.60
0.2
0.4
0.6
VxVy
Vz0
500
1000
1500
2000
2500
3000
Vz Vy Vx
(ms)
(b) E
nd o
f P w
ave
(a) B
egin
ning
of P
Wav
e (c
) QR
S co
mpl
ex
Bod
y
Hea
d
Tail
63
Figu
re 5
.1 F
ram
es fr
om d
ynam
ic V
CG
repr
esen
tatio
n of
a p
atie
nt w
ith m
yoca
rdia
l inf
arct
ion,
ant
erio
r
-0.2
00.
20.
40
0.2
0.4
0.60
0.2
0.4
0.6
VxVy
Vz0
500
1000
1500
2000
2500
3000
Vz Vy Vx
(ms)
-0.2
00.
20.
40
0.2
0.4
0.60
0.2
0.4
0.6
Vx
Vy
Vz0
500
1000
1500
2000
2500
3000
Vz Vy Vx
(ms)
-0.2
00.
20.
40
0.2
0.4
0.60
0.2
0.4
0.6
Vx
Vy
Vz0
500
1000
1500
2000
2500
3000
Vz Vy Vx
(ms)
(f)
Seco
nd Q
RS
com
plex
(e
) End
of T
wav
e (d
) Mid
dle
of T
wav
e
64
Therefore, this explicitly real time dynamic VCG representation makes it easier for
doctors to utilize it with prior interpretation expertise and experiences of time-based
ECG. As shown in Figure 5.1 (c), this representation consists of three components: head
(green), body (red), and tail (blue). Head gives the current position of the cardiac vector.
Body records a short history of the cardiac movement which clearly indicates where the
current vector is from. It avoids the confusion regarding which heart activity loop the
current cardiac vector belongs to because those loops usually have some intersections in
the isoelectric point. Tail provides the full history from the start of the recording time that
captures the full topological shapes of the VCG attractor. Figure 5.1 (a) bottom VCG
trajectory shows the beginning of P wave which is the start of atrial depolarization after
the SA node excitation. This cardiac activity is also indicated from the projections of
cardiac vector on the X, Y, Z Cartesian axes as shown in the Figure 5.1 (a) top plot. In
Figure 5.1 (b), we can see the complete P wave loop and Q wave onset (interventricular
septum depolarization). When it comes to Figure 5.1 (c), the QRS loop is presented to
manifest the ventricular depolarization activities. It is followed by ventricular
repolarization shown as T wave loop in Figure 5.1 (d) (e). Figure 5.1 exhibits the next
heart activity cycle. Figure 5.1 clearly shows that the P, QRS, and T waves can be easily
located in the VCG attractor with respect to time.
5.2.2 Lag reconstructed spatiotemporal ECG attractor
In many real life cases, only a few measurements (e.g., lead I ECG signal) are
available in lieu of the complete measurements of three cardiac vector components
recorded for a given time period. Takens embedding theorem [18] states that many
individual measurements contain adequate information to reconstruct most of the system
65
attractor dynamics because of the high dynamic coupling existing among the real world
system. Therefore, an equivalent cardiac vector space (attractor) can be reconstructed
from the delayed coordinates of the only measurement y(t) as
v(ti) =[y(ti), y(ti+�), y(ti+2�)] (Eq 5-2)
where � is the time delay. The optimal time delay � is selected to minimize mutual
information function M(�), defined as
( , )( ) ( , ) log( ) ( )
�� ��
�t
p t tM p t t dtp t p t
(Eq 5-3)
where p(t,�) is the joint density function, and p(t) and p(t+�) are marginal density
functions of y(t) and y(t+�), respectively [21].
As shown in Figure 5.2, animation of the lag reconstructed EKG attractor can also
provide the similar spatiotemporal information as the VCG attractor described in Section
5.2.1.
66
500
1000
1500
2000
2500
3000
-0.20
0.2
0.4
0.6
Potentials
(ms)
00.
20.
40
0.2
0.4
0
0.2
0.4
x(t)
x(t+�)
x(t+2�)
500
1000
1500
2000
2500
3000
-0.20
0.2
0.4
0.6
Potentials(m
s)
00.
20.
40
0.2
0.4
0
0.2
0.4
x(t)
x(t+�)
x(t+2�)50
010
0015
0020
0025
0030
00-0
.20
0.2
0.4
0.6
Potentials
(ms)
00.
20.
40
0.2
0.4
0
0.2
0.4
x(t)
x(t+�)
x(t+2�)
(b) E
nd o
f P w
ave
(a) B
egin
ning
of P
Wav
e (c
) Sta
rt of
QR
S co
mpl
ex
67
Figu
re 5
.2 F
ram
es fr
om la
g re
cons
truct
ed d
ynam
ic E
KG
attr
acto
r rep
rese
ntat
ion
for a
nor
mal
subj
ect
500
1000
1500
2000
2500
3000
-0.20
0.2
0.4
0.6
Potentials
(ms)
00.
20.
40
0.2
0.4
0
0.2
0.4
x(t)
x(t+�)
x(t+2�)
500
1000
1500
2000
2500
3000
-0.20
0.2
0.4
0.6
Potentials(m
s)
00.
20.
40
0.2
0.4
0
0.2
0.4
x(t)
x(t+�)
x(t+2�)
500
1000
1500
2000
2500
3000
-0.20
0.2
0.4
0.6
Potentials
(ms)
00.
20.
40
0.2
0.4
0
0.2
0.4
x(t)
x(t+�)
x(t+2�)
(e) D
urin
g T
wav
e (d
) Dur
ing
QR
S co
mpl
ex
(f) E
nd o
f T W
ave
68
5.2.3 Color coding of spatiotemporal VCG representation
Color coding scheme can be used to incorporate additional dynamical attributes of
the VCG attractor, such as curvature, “speed" (dv = /v//t, and /v=||v(t)-v(t+1)||) and
phase angle. The estimate of the curvature can be made by taking the cross product
between the two vectors from the current point to the points a certain distance on either
side. The phase angles of the cardiac vector are calculated using Eq 5-4.
2 2 2
cos x
x y z
vv v v
(
,2 2 2
cos y
x y z
v
v v v!
, and
2 2 2cos
z
x y z
v
v v v� (Eq 5-4)
The magnitude of the color indicators, such as curvature, speed, phase angles are
mapped into a color scale. Thus, the color coding scheme can be plotted in real time and
provide the fifth dimension of VCG besides X, Y, Z, and time scale. As shown in Figure
5.3, the color coded animated VCG representation incorporates the extra cardiac vector
movement information and facilitates doctor’s interpretation of valuable spatiotemporal
patterns associated with certain cardiological diseases. Figure 5.3 bottom plot shows the
cardiac vector potential variation speed plot, and the middle VCG plot displays every
vector’s speed in one specific color corresponding to the color bar listed in the right of
the middle VCG plot. The same color bar range is applied to all the color coded dynamic
VCG representation so as to see the differences among those colorful plots. As illustrated
by the color VCG attractor in Figure 5.3, the cardiac vector in QRS loop moves fastest, T
wave is the second, P wave is the third, and the isoelectric points are the slowest.
69
Patient266/s0502_re: Healthy control RR: 696.03±4.67ms, QT: 435.62±5.17ms
Figure 5.3 Color coded dynamic VCG representation plot of a healthy control subject
-0.2 0 0.2 0.4 0.6 0.800.2
0.40.6
-0.2
0
0.2
0.4
0.6
VxVy
Vz
500 1000 1500 2000 2500 3000
Vz
V
y
V
x
(ms)
0 500 1000 1500 2000 2500 3000
0.005
0.01
0.015
0.02
0.025
Pot
entia
l var
iatio
n ra
te (m
v/m
s)
(ms)
QRS Wave
T Wave Vx – positive Vy – positive Vz - negative
P Wave
Potential variation rate (mv/ms)
70
5.3 Results and discussion
Many researchers have studied the pathological patterns in the VCG signals and
demonstrated the advantages of VCG in diagnosing certain diseases [9-20]. Statistical
transformation studies showed the equivalence of 12-lead EKG and 3-lead VCG.
Therefore, an effective VCG representation can provide same heart characteristics as
acquired from the conventional ECG signals. The useful heart characteristics to be
determined include heart rate, rhythm, electrical axis, and other pathology patterns. As
shown in Figure 5.3, the healthy control VCG plot presents the normal heart rhythm as P,
QRS, and T loops. The peak of T wave loop is found to be in the octant (XYZ: ++-) for
most of the healthy control cases. Statistical analysis shows that only 6 out of 80 healthy
recordings in PTB database flee from the octant (XYZ: ++-). But for the peak of T wave
loop in PTB myocardial infarction cases, only 31.79% (117/368) are found to stay in the
octant (XYZ: ++-) and the rest 68.21% of myocardial infarction cases get into the other
octants. The speed color coding scheme manifest the real time movement velocity of the
cardiac vector, which is rarely able to be observed on a paper. The color of QRS loop is
continuous in the healthy control case. For instance, Figure 5.3 shows a consistent light
green color, which is usually not the case for some disease subjects. It may also be noted
that the more red color shown in QRS loop, the higher R peak is. In addition, Q wave and
S wave are inside the QRS loop in the vectorcardiogram and close to the P loop and T
loop. Since this dynamic VCG representation is printed on a paper statically, it is not
straightforward to indentify which side of QRS loop is Q wave. It is well known that Q
wave is ahead of R wave and S wave. Therefore Q wave can be easily located on the
monitor from real time motion of dynamic VCG representation. Here, the Q wave is
71
marked in Figure 5.3 and the normal subject is found to not have a significant Q wave
because the amplitude of Q wave does not noticeably incline to the negative direction of
X, Y, or Z directions.
But P wave loop is found missing and abnormal in the case of atrial fibrillation
VCG plot and fibrillation waves take the place of P wave (see Figure 5.4). It is also found
that the attractor of atrial fibrillation patient is approximately parallel to the X-Y plane,
which indicates a shift in the heart electrical axis. Figure 5.4 shows the color coded
dynamic VCG representation for a dysrhythmia and atrial fibrillation patient. P wave is
ahead of QRS loop. Before the VCG trajectory goes to the QRS wave, it is found to be
trapped in the anarchy P area. The speed of cardiac vector movement is displayed as the
darkest blue color (minimal potential variation rate), which demonstrated clearly
abnormal excitations from SA node to Atrial. The presence of chaotic fibrillation wave
before QRS loop discloses that the atrial fibrillation case has limited capability to harness
Atrial and SA node. The time elapsed for the completion of one VCG cycle (PQRST)
provides the RR intervals, and it can be calculated for the study of heart rate variability
using Poincare sectioning of VCG trajectory. In Figure 5.4, the RR interval is found to be
longer than the normal case. It may also be noted that the color change in QRS loop is not
as consistent as the normal case in Figure 5.3, which indicates abnormalities in
ventricular depolarization.
72
Patient157\s0338lre: Dysrhythmia and atrial fibrillation RR: 1571.20±108.34ms, QT: 476.50±3.64ms
Figure 5.4 Color coded dynamic VCG representation plot of a patient with dysrhythmia
and atrial fibrillation
-0.1 0 0.1 0.2 0.3 0.4 0.5-0.4-0.20
-0.2
0
0.2
0.4
0.6
VxVy
Vz
500 1000 1500 2000 2500 3000
Vz
V
y
V
x
(ms)
0 500 1000 1500 2000 2500 3000
0.005
0.01
0.015
0.02
0.025
0.03
Pot
entia
l var
iatio
n ra
te (m
v/m
s)
(ms)
FibrillationWave
Potential variation rate (mv/ms)
73
Further, the pathology patterns, such as significant Q wave, inverted T wave for
myocardial infarction patients can also be indentified from the VCG plot. A large Q wave
towards the negative direction along the Y-axis can be seen in Figure 5.5 because the Y-
axis magnitude of starting moment of the QRS loop is significantly larger than the Y-axis
magnitude of R peak. The T wave octant is also found to be shifted to X-positive, Y-
negative and Z-positive, which indicates the inverted T wave. A significant Q wave can
also be located from Figure 5.5 Vy direction. It may be noted that the minimal negative
direction of Vy is less than – 0.2, but the maximal positive direction is only 0.2. Q wave
is the starting of QRS loop, which can be easily reserved from dynamic VCG
representation. Therefore Q wave is found to be beyond -0.2 and Q/R ratio is greater than
1 in the Vy direction. Such a significant Q wave is definitely a typical sign of myocardial
infarction. The discontinuous color variations in QRS loop also indicate the ‘W’ patterns
in ventricular depolarization, which are pathological patterns for some diseases.
74
Patient001/s0010_re: Myocardial infarction, infero-latera RR: 732.16±8.75ms, QT: 430.16±3.12ms
Figure 5.5 Color coded dynamic VCG representation plot of a patient with myocardial
infarction, infero-latera
-0.3 -0.2 -0.1 0 0.1 0.2 0.3-0.20
0.2
-0.2
0
0.2
0.4
VxVy
Vz
500 1000 1500 2000 2500 3000
Vz
V
y
V
x
(ms)
0 500 1000 1500 2000 2500 3000
0.005
0.01
0.015
0.02
Pot
entia
l var
iatio
n ra
te (m
v/m
s)
(ms)
Inverted T Wave
Potential variation rate (mv/ms)
SignificantQ Wave
Inverted T Wave Vx – positive Vy – negative Vz - positive
75
Figure 5.6 clearly shows a significant Q wave along X and Y axes. It may also be
noted that Q and S wave are significantly larger than R wave. Thus, this tiny R wave
results in a deep valley in the Figure 5.6 bottom cardiac vector potential variation speed
plot. It is also revealed more obviously in the Figure 5.6 middle VCG plot. This
myocardial infarction case exhibits an even worse significant Q wave situation than
Figure 5.5. A significant Q wave can be located from both Vx and Vy directions. We can
find that the minimal negative direction of Vy is less than – 0.6, but the maximal positive
direction is less than 0.2. The minimal negative direction of Vx is less than – 0.2, but the
maximal positive direction is only 0.05. Q wave is located in the dynamic VCG
representation (see Figure 5.6) and Q/R ratio is found to be greater than 1 in both Vx and
Vy directions. The discontinuous color variations in QRS loop are shown as light blue –
dark blue – light blue. It indicates the ‘W’ patterns in the QRS wave of this patient, which
is resulted from the tiny R wave compared to Q and S waves. It may also be noted that
cardiac vector speed is showing to be dark blue in both P and T wave areas. Slow
movement in these two locations also represents the SA excitation and ventricular
repolarization abnormalities.
76
Patient063/s0214lre: Myocardial infarction, antero-lateral RR: 822.96±101.62ms, QT: 401.30±31.38ms
Figure 5.6 Color coded dynamic VCG representation plot of a patient with myocardial infarction, antero-lateral
-0.2 -0.15 -0.1 -0.05 0 0.05-0.6-0.4
-0.20
0
0.2
0.4
VxVy
Vz
500 1000 1500 2000 2500 3000
Vz
V
y
V
x
(ms)
0 500 1000 1500 2000 2500 3000
0.005
0.01
0.015
0.02
Pot
entia
l var
iatio
n ra
te (m
v/m
s)
(ms)
SignificantQ Wave
SignificantQ Wave
Potential variation rate (mv/ms)
Deep valley from Tiny
R peak
Tiny R peak
77
As shown in Figure 5.7, two R peaks can be found in the plot which are
designated as “M waves” and are one of the typical pathological patterns for the bundle
branch block patient. The discontinuous color variations in QRS loop are shown as
yellow - light blue – dark red. It indicates the ‘M’ patterns in the R wave of this patient,
which is resulted from the disease of bundle branch block. The ‘M’ wave pattern shows
that both ventricles are not depolarized simultaneously. The delay in the blocked bundle
branch allows the unblocked ventricle to begin depolarizing before the blocked ventricle.
This kind of slightly later effect in one ventricle produces pathological ‘M’ wave and
widen QRS loop. It may also be noted that the dark red color appears in this VCG
trajectory. Since the same color scale is employed, and Figure 5.3, Figure 5.4, Figure 5.5,
Figure 5.6 do not have red color in the VCG attractor, it is also concluded that R peak
amplitude is the largest in the bundle branch block subject.
78
Patient208/s0429_re: Bundle branch block RR: 745.48±7.77ms, QT: 465.15±14.74ms
Figure 5.7 Color coded dynamic VCG representation plot of a patient with bundle branch
block
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2-0.4-0.200.20.4
-0.5
0
0.5
1
1.5
2
VxVy
Vz
500 1000 1500 2000 2500 3000
Vz
V
y
V
x
(ms)
0 500 1000 1500 2000 2500 3000
0.01
0.02
0.03
0.04
0.05
Pot
entia
l var
iatio
n ra
te (m
v/m
s)
(ms)
R’
R
Potential variation rate (mv/ms)
79
From the VCG attractor comparisons among healthy control (HC), myocardial infarction
(MI), atrial fibrillation (AF), and bundle branch block, it can be shown that a rotatable real time
dynamic VCG representation can capture these important heart characteristics as reported from
the 12-lead ECG plot. It is convenient to find the pathological patterns of various cardiovascular
disorders from the dynamic VCG representation. One perspective of view or static representation
on a paper will pose additional difficulties to the interpretations of VCG trajectory. The
incorporation of time and color in dynamic VCG representation will greatly facilitate and assist
the inspection of VCG attractor.
Moreover, 3-lead VCG signals are better options for computer automated analysis and
process than the 12-lead ECG because of the dimension reduction. From the rotated dynamic
VCG representation, it is also found that the spatial distributions of VCG trajectory are
remarkably different between myocardial infarction and healthy control. In our chapter VIII,
VCG features, namely, max Q vector magnitude, max Q vector octant, max R vector magnitude,
max R vector octant, max T axis magnitude, max T vector octant, heart rate, and ST segment
integration area etc., are extracted to identify the myocardial infarction from healthy control. T
wave morphology related information was statistically found to be the most important features,
and the classification accuracy was as high as 97.28% for myocardial infarction and 93.75% for
healthy control cases in the PTB database.
5.4 Concluding remarks
In this present investigation, we have made an attempt to represent the VCG signals in 3D
space in real time by viewing the signals on a computer monitor instead of as a static picture
which can at best record 3 dimensions. This approach overcomes the drawbacks of conventional
static VCG representation and provides the spatiotemporal information on the heart dynamics.
80
The alternative lag reconstructed heart attractor animation from nonlinear dynamics principles
can address the sometime difficult situations for the study of cardiac vector when there is only
one heart monitoring signal available. The VCG representation can also be color coded for
spatiotemporal representation with the curvature, velocity, and phase angle etc. of the vector loops
for valuable information about the heart’s dynamic activities. It is believed that the proposed five
dimension dynamic VCG representation, namely, X, Y, Z, time, and color, can significantly
enhance the interpretability of the vectorcardiogram and assist in the identification of the
underlying spatiotemporal heart dynamics.
5.5 References
[1] J. Malmivuo and R. Plonsey, Bioelectromagnetism: Principles and Applications of
Bioelectric and Biomagnetic Fields. USA: Oxford University Press, July 18, 1995.
[2] A. Goldberger, L. Amaral, L. Glass, J. Hausdorff, P. Ivanov, R. Mark, J. Mietus, G.
Moody, C. Peng, and H. Stanley, "PhysioBank, PhysioToolkit, and PhysioNet:
Components of a New Research Resource for Complex Physiologic Signals," Circulation
101, vol. 23, pp. e215-e220, 2000 (June 13).
[3] S. S. Barold, "Willem Einthoven and the Birth of Clinical Electrocardiography a Hundred
Years Ago," Cardiac Electrophysiology Review, vol. 7, pp. 99-104, 2003.
[4] G. E. Dower and H. B. Machado, "XYZ data interpreted by a 12-lead computer program
using the derived electrocardiogram," Journal of electrocardiology, vol. 12, pp. 249-261,
1979.
[5] G. E. Dower, H. B. Machado, and J. A. Osborne, "On deriving the electrocardiogram
from vectorcardiographic leads," vol. 3, p. 87, 1980.
[6] G. E. Dower, A. Yakush, S. B. Nazzal, R. V. Jutzy, and C. E. Ruiz, "Deriving the 12-lead
electrocardiogram from four (EASI) electrodes," Journal of electrocardiology, vol. 21,
pp. S182-S187, 1988.
81
[7] V. P. Hubert, "Advantages of Three Lead Cardiographic Recordings," Annals of the New
York Academy of Sciences, vol. 126, pp. 873-881, 1965.
[8] P. W. Macfarlane and T. D. V. Lawrie, Comprehensive Electrocardiology: Theory and
Practice in Health and Disease, Chapter 11 Lead Systems. New York, NY: Pergamon
Press, 1989.
[9] A. Benchimol, F. Reich, and K. B. Desser, "Comparison of the electrocardiogram and
vectorcardiogram for the diagnosis of left atrial enlargement," Journal of
electrocardiology, vol. 9, pp. 215-218, 1976.
[10] I. Black, "The electrocardiogram and vectorcardiogram in congenital heart disease," The
Journal of Pediatrics, vol. 69, pp. 846-716, 1966.
[11] B. W. McCall, A. G. Wallace, and E. H. Estes, "Characteristics of the normal
vectorcardiogram recorded with the Frank lead system," The American Journal of
Cardiology, vol. 10, pp. 514-524, 1962.
[12] J. Ng, A. V. Sahakian, W. G. Fisher, and S. Swiryn, "Surface ECG Vector Characteristics
of Organized and Disorganized Atrial Activity During Atrial Fibrillation," Journal of
Electrocardiology, vol. 37, pp. 91-98, 2004.
[13] C. Olson and R. A. Wamer, "The quantitative 3-diemnsional vectorcardiogram," Journal
of electrocardiology, vol. 33, pp. 176-S74, 2000.
[14] D. M. Schreck, J. D. Frank, E. R. Malinowski, L. E. Thielen, A. F. Bhatti, A. R. Rivera,
and V. J. Tricarico, "Mathematical modeling of the electrocardiogram and
vectorcardiogram using factor analysis," Annals of Emergency Medicine, vol. 23, pp.
617-76, 1994.
[15] E. Simonson, N. Tuna, N. Okamoto, and H. Toshima, "Diagnostic accuracy of the
vectorcardiogram and electrocardiogram : A cooperative study," The American Journal
of Cardiology, vol. 17, pp. 829-878, 1966.
82
[16] K. K. Talwar, S. Radhakrishnan, V. Hariharan, and M. L. Bhatia, "Spatial
vectorcardiogram in acute inferior wall myocardial infarction: its utility in identification
of patients prone to complete heart block," International Journal of Cardiology, vol. 24,
pp. 289-292, 1989.
[17] A. van Oosterom, Z. Ihara, V. Jacquemet, and R. Hoekema, "Vectorcardiographic lead
systems for the characterization of atrial fibrillation," Journal of electrocardiology, 2006.
[18] J. L. Willems, E. Lesaffre, and J. Pardaens, "Comparison of the classification ability of
the electrocardiogram and vectorcardiogram," The American Journal of Cardiology, vol.
59, pp. 119-124, 1987.
[19] F. N. Wilson and F. D. Johnston, "The vectorcardiogram," American Heart Journal, vol.
16, pp. 14-28, 1938.
[20] A. C. Witham, "Quantitation of the vectorcardiogram," American Heart Journal, vol. 72,
pp. 284-286, 1966.
[21] H. Yang, S. T. Bukkapatnam, and R. Komanduri, "Nonlinear adaptive wavelet analysis of
electrocardiogram signals," Physical Review E (Statistical, Nonlinear, and Soft Matter
Physics), vol. 76, p. 026214, 2007.
PART II
FEATURE EXTRACTION AND CLASSIFICATION
83
CHAPTER VI
ATRIAL FIBRILLATION STATE CLASSIFICATION
6.1 Introduction
Atrial fibrillation (AF) is the most common form of arrhythmia, affecting some 2.2
million Americans annually. AF can sustain indefinitely, since the ventricles continues to
perform the essential function of blood circulation. During AF, the heart's two small
upper chambers (the atria) (see Figure 2.2) quiver instead of beating effectively. The risks
of sustained atrial fibrillation are nevertheless serious, and include strokes and
myocardial infarctions caused by the formation of blood clots with stagnant volumes in
the atria [1]. About 15% of strokes occur in patients are traced to AF. The likelihood of
developing AF increases with age with about 3-5% of patients over 65 have atrial
fibrillation and is more for men than for women.
In surface electrocardiogram (ECG) from patients with AF, regular electrical
impulses of the sinoatrial node (P wave) as shown in Figure 2.2 are replaced by
disorganized, rapid electrical impulses (Fibrillation or F wave) which result in irregular
heart beats [2-5]. The atrial electrical activity is very rapid (approximately 400 to 700
beats/min), but each electrical impulse results in the depolarization of only a small islet of
atrial myocardium rather than the entire atrium. A typical ECG in AF shows a rapid
irregular tachycardia in which recognizable P waves are absent. QRS complexes are
generally normal, and the ventricular rate in patients with untreated AF generally ranges
84
between 150 and 220 beats/min. However, in elderly patients, ventricular rates in
untreated AF are typically slower [1].
Evidence shows that spontaneously terminating (paroxysmal) AF is a precursor to the
development of sustained AF [6]. Subtle changes in rhythm during the final minutes or
seconds of such episodes may lead to (or predict) termination of AF. Improved
understanding of the mechanisms of spontaneous termination of AF may lead to
improvements in the treatment of sustained AF. The remainder of chapter 6 is organized
as follows: Section 6.2 gives a brief background of the 2004 PhysioNet challenge for
classification of AF states; Section 6.3 contains the study of feature extraction through
statistical analysis and QRST subtraction; AF state classification using classification and
regression tree (CART) model is discussed in Section 6.4; and Sections 6.5 discuss the
results and future improvements.
6.2 Background
In this chapter, the ECG data posted on the world-wide web (www) from the 2004
PhysioNet challenge “Spontaneous Termination of Atrial Fibrillation” is utilized. In this
contest, classification needs to be made among the following three categories of AF
patients test signals:
1) Group N: Non-terminating AF (defined as AF that was not observed to have
terminated for the duration of the long-term recording, at least an hour following
the segment).
2) Group S: Soon to be terminating (AF that terminates one minute after the end of
the record).
85
3) Group T: Terminating immediately (AF terminating within one second after the
end of the record).
In all, 80 recordings of AF from 60 different subjects were made available in the
database [7]. Each record is a one-minute long segment of ECG signals sampled from
two channels at 128 Hz sampling rate. The data is divided into a learning set and two test
sets. The learning set contains 30 records in all, with 10 records in each of the three
groups. The learning set records were obtained from 20 different subjects (10 from group
N and 10 from group S/T). Test set A contains 30 records from 30 subjects. About half of
these records belong to group N and the rest to group T. The first goal of the challenge,
referred to as Event A, is to determine which records in test set A belong to group T and
which records to group N. Test set B contained 20 records, 2 from each of 10 subjects
(none represented in the learning set or in test set A). One record of each pair belongs to
group S and the other to group T. The second goal, referred to as Event B, is to identify
which records belong to group T and which records belong to S. It may be noted that the
data provided was sparse (as shown in Table 6.1) making it all the more difficult and
challenging. Some 20 groups participated in the challenge and submitted their results to
an autoscorer. Each team was allowed up to five attempts in each event, and only the best
score received in each event was used to determine the final rankings. Final results for the
2004 PhysioNet Challenge are summarized in Table 6.2.
Table 6.1 Distribution of 80 recordings among learning sets and test sets
Group N Group S Group T Learning set n01, n02, ... n10 s01, s02, ... s10 t01, t02, ... t10
Test set A 15 records - 15 records Test set B - 10 records 10 records
86
Tab
le 6
.2 S
umm
ary
of fi
nal r
esul
ts o
f the
200
4 Ph
ysio
Net
Cha
lleng
e
Ref
eren
ce
Met
hod
Res
ults
R
epor
ted
Perf
orm
ance
R
emar
ks
Eve
nt I
(N&
T)
Eve
nt II
(S&
T)
Petru
tiu [8
]
Ban
dpas
s filt
erin
g, Q
RST
subt
ract
ion,
FFT
to
resi
dual
sign
al, o
vera
ll pe
ak fr
eque
ncy,
10
s pe
ak fr
eque
ncy,
1 s
peak
freq
uenc
y an
d th
e tre
nd o
f pea
k fr
eque
ncy
chan
ge
Trai
ning
: 20/
20
Test
ing:
29/
30
Trai
ning
: 20/
20
Test
ing:
20/
20
Even
t I:9
7.5%
Ev
ent I
I:100
%
Perf
orm
ance
ave
rage
d ov
er
the
test
cas
es o
nly
Hay
n [9
]
QR
ST su
btra
ctio
n &
shor
t tim
e Fo
urie
r tra
nsfo
rm to
the
AA
sign
al, c
lass
ifica
tion
with
fe
atur
es: p
eak
AF
freq
uenc
y an
d m
ean
RR
in
terv
al
Trai
ning
: 20/
20
Test
ing:
28/
30
Trai
ning
: 20/
20
Test
ing:
16/
20
Even
t I:9
6%
Even
t II:9
0%
Ave
rage
d ov
er tr
aini
ng a
nd
test
ing
case
s
Can
tini [
10]
Sepa
rate
AA
from
VA
usi
ng: (
1) a
dapt
ive
filte
ring
QR
ST su
btra
ctio
n (2
) mea
n be
at
subt
ract
ion,
feat
ures
: RR
inte
rval
and
pea
k at
rial
freq
uenc
y (e
vent
I), p
eak
freq
uenc
y of
last
two
seco
nds c
hang
e (e
vent
II)
Trai
ning
: 19/
20
Test
ing:
27/
30
Trai
ning
: 16/
20
Test
ing:
18/
20
Even
t I:9
2%
Even
t II:8
5%
Ave
rage
d ov
er tr
aini
ng a
nd
test
ing
case
s
Lem
ay [1
1]
Hig
hpas
s filt
erin
g, fe
atur
es: s
kew
ness
of A
A
segm
ents
, mea
n of
RR
inte
rval
diff
eren
ce in
ten
seco
nds,
F w
ave
peak
inte
rval
s, at
rial a
ctiv
ity
ampl
itude
, tot
al p
ower
bel
ow 0
.6H
z on
bot
h le
ads,
cros
s-co
varia
nce
valu
es fo
r tw
o le
ads,
estim
ated
spec
tral d
ensi
ty a
bove
20H
z ac
ross
A
A se
gmen
ts o
n bo
th le
ads,
linea
r mod
elin
g
Trai
ning
: 20/
20
Test
ing:
20/
30
Trai
ning
: 20/
20
Test
ing:
12/
20
Even
t I:8
0%
Even
t II:8
0%
Ave
rage
d ov
er tr
aini
ng a
nd
test
ing
case
s
Cas
tells
[12]
Pr
inci
pal C
ompo
nent
Ana
lysi
s (PC
A),
PQR
ST
cycl
e Tr
aini
ng: 2
0/20
Te
stin
g: 2
7/30
N
/A
Even
t I: 9
4%
Even
t II:
N/A
A
vera
ged
over
trai
ning
and
te
stin
g ca
ses
Nils
son
[13]
Sp
atio
-tem
pora
l QR
ST su
btra
ctio
n an
d Ti
me-
Freq
uenc
y A
naly
sis
Trai
ning
: 19/
20
Test
ing:
27/
30
Trai
ning
: 16/
20
Test
ing:
14/
20
Even
t I: 9
2%
Even
t II:
75%
A
vera
ged
over
trai
ning
and
te
stin
g ca
ses
87
Most successful approaches followed the procedure of filtering the noise and artifacts
[8, 10, 11, 14], QRST subtraction [15-20], frequency spectrum analysis of the resulting
filtrate (which predominantly captures atrial activity) [8, 21-26], classification [11, 24,
27] and so on. Six groups developed methods that yielded 90% or more accurate results
for event A (N vs. T classification). Studies show that atrial activity slows down and
regularizes prior to self-termination of AF, mechanisms of atrial activity are reflected
through surface ECG [1-3, 5, 26, 28, 29]. However, event B (S vs. T classification)
seemed to be more difficult, since the accuracies ranged from 60-90% for most cases. It
is commendable that the team from Northwestern University [8] was able to arrive at the
highest classification accuracy of 97% for the N vs.T and 100% for S vs. T. One major
finding of this challenge contest is that spontaneously terminating AF can be
discriminated accurately from sustained AF by analysis of surface ECG signal segments
sampled over a period of one minute or less [6].
6.3 Feature extraction
6.3.1 Statistical Descriptors and Principal Component Analysis (PCA) Various signal processing algorithms have been developed to detect the QRS
complex and the characteristic points of ECG spectra [14, 30-41]. This stage is crucial in
basic ECG monitoring systems and important for cardiac events classification. Figure 6.1
shows an intermediate result of an automated feature extraction algorithm developed here
depicting the successful identification of peaks and endpoints of various waves of a
representative ECG signal. All signals were first passed through a bandpass filter to
capture a clean signal whose frequency range is between 0.5 and 60 Hz before subjecting
them to wavelet transform as discussed in the foregoing. Pre-filtering minimizes the
88
influence of artifacts, respiration, baseline shift and other spurious signal components, so
that the features extracted are more likely to accurately track variations in the AF states.
Figure 6.1 Automated identification of peaks and endpoints of various waveforms in an ECG signal using an automated feature extraction routine
Wavelets are attractive for ECG feature extraction because the spatio-temporal
features of an ECG signal are presented over multiple scales through the use of a wavelet
transform. Consequently, the QRS complex can be distinguished from high P and T
waves, noise, baseline drift, and other artifacts. The relation between the characteristic
points of ECG signals and those of modulus maximum pairs of its wavelet transforms are
almost straightforward to establish. For example, Figure 6.2 shows a representative
scalogram (time-scale representation) obtained from a continuous wavelet transform
(CWT) of an ECG signal segment shown in Figure 6.2. The brighter (white) portions
refer to the time-scale segments in the scalogram that contain large magnitudes of
wavelet coefficients and the dark portions identify the segments having small wavelet
coefficients. Scale 2 of the scalogram clearly demarcates the time locations of R peaks.
0 50 100 150 200 250
-0.2
0
0.2
0.4
0.6
0.8
EKG Signal in Time Domain
Time Index
Mill
ivol
ts
P
R
QS
T
89
The information on R-peak locations gets diffused as one moves up on the scales, as
evidenced by progressive smearing of the white bands. However, around scale 7,
delineations between R, T, and P waves become somewhat apparent. However, for scales
>7, these bands diffuse out in time to a point where the delineations are no longer clear.
Scale 2 of a scalogram has been used to locate the peaks of the R waves, and thus
determine, for each beat, the heart rate (HR) and heart rate variability (HRV). Scale 7
coefficients were used to determine the peaks of the P and T waves as well as the onset
and offset points of each wave.
Figure 6.2 Time-scale representation (scalogram) of a representative ECG signal
200 400 600 800 1000
A small scale corresponds to a high frequency. The R wave frequency component is about a scale of 2 where its CWT coefficients are large (white).
A high scale corresponds to a low frequency. P and T wave frequency components are expected around a scale of 7.
Continuous Wavelet Transform to ECG Signal in Different Scales
Time index
1
2
3
4
5
6
7
8
scal
es
0 200 400 600 800 1000-2
0
2ECG signal in time domain
90
The feature sets extracted from large arrays of signals has been further tightened,
and made more sensitive to the relevant variations in underlying dynamics through the
use of decomposition methods, such as principal component analysis (PCA) and
independent component analysis (ICA). PCA techniques [42] are widely used to reduce
the dimension of the input (feature) vectors in situations where the dimension of the input
vector is large, but the components of the vectors are highly correlated (redundant). PCA
orthogonalizes the components of the input vectors (so that they are uncorrelated with
each other), and orders the resulting orthogonal components (principal components) so
that those with the largest variation are considered first, and eliminates those components
that contribute the least to the variation in the data set.
6.3.2 QRST subtraction During atrial fibrillation, the P waves are absent and are replaced by rapid (usually of
smaller amplitude) fibrillation waves (or F-waves) of varying durations and
morphologies. The QRST complex (which represents ventricular activity) dominates the
spectral content of ECG waves during AF, limiting the ability to detect various AF
events. Therefore, the ECG signal components capturing the ventricular activity (i.e.,
QRST complex) should be subtracted from the surface ECG for more effective diagnosis
of AF events [15-20, 43]. In this chapter, a wavelet representation is used to identify time
segments and scales where much of the ECG signal content emerges from various
ventricular activities, and apply and threshold such that the filtrate contains mostly
ventricular components and the residue contains atrial components. The rationale for this
approach lies in the fact that the wavelet transform coefficients for ventricular activity are
much larger than those for atrial activity, particularly during active AF. Results from this
91
cancellation approach are summarized in Figure 6.3 and Figure 6.4. It is evident that
much of the QRST information in the ECG has been culled out through the application of
the thresholding filter. The frequency domain plot of the QRST cancelled ECG (Figure
6.4, bottom) clearly delineates the spectral content of the waves of atrial origin, which
were not evident from the examination of the frequency spectrum of the original ECG
(Figure 6.4, top). The frequency and amplitudes of these atrial ECG components serve as
additional features for AF state classification.
Figure 6.3 Summary of QRST subtraction results for a representative ECG signal (from a T-case) showing the original signal (top), filtrate from thresholding containing
predominantly ventricular components (middle) and residue from thresholding containing predominantly atrial and AF waves (bottom)
0 200 400 600 800 1000 1200 1400 1600 1800 2000-1
0
1
Set T/t1 Time Domain Signal
0 200 400 600 800 1000 1200 1400 1600 1800 2000-1
0
1
Ventricular Activity Signal (QRST)
0 200 400 600 800 1000 1200 1400 1600 1800 2000-0.2
0
0.2
Atrial Activity Signal ( Fibrillation Wave)
92
Figure 6.4 Frequency spectra of a representative ECG signal before and after QRST
subtraction
6.4 AF state classification using CART model
The CART analysis uses decision rules learned from the pattern (clustering) of
features in a training data set to classify different groups of objects. This approach can be
quite effective for creating clinical decision rules that combine physiological
knowledgebase with signal analysis [27].
The application of CART analysis to detect AF events (N, S, and T) consists of
selecting the features to serve as discriminatory variables and build a decision tree
recursively. The CART tree is built using recursive splitting of the feature space defined
by 3 extracted input features based on one target variable that takes one of the following
3 values: N, S, or T.
0 5 10 15 200
0.1
0.2
0.3
0.4Set T/t1 Frequency Spectrum before QRST cancellation
Frequency (Hz)
Am
plitu
de
0 5 10 15 200
0.01
0.02
0.03Set T/t1 Frequency Spectrum after QRST cancellation
Frequency (Hz)
Am
plitu
de Atrial relatedactivity
Primary pulse rate
T wave
R peak
93
(a)
(b)
Figure 6.5 (a) Peak frequencies of various N and T cases, (b) Peak frequencies of the
first two (First 2) and last two (Last 2) beats of N and T ECG signals
For N vs T classification, subtle changes in the rhythm during the final seconds of the
ECG traces of such episodes may indicate termination of AF [25]. Therefore, ECG
features that capture the initial (first 2 seconds) and final moments (last 2 seconds) of AF
signal may serve as discriminatory variables for identifying AF termination. Specifically,
the following 3 features were extracted from the ECG signals after QRST subtraction:
N T4
4.5
5
5.5
6
6.5
7
7.5
8N and T Learning Dataset Overall Peak Frequency
Ove
rall
Peak
Fre
quen
cy
T cases: opf < 5.13
N cases: opf > 6.91
OverlappingArea
First 2 Last 2 Fisrt 2 Last 23
4
5
6
7
8
9N and T Learning Dataset Two Second Peak Frequency
Two
Sec
ond
Peak
Fre
quen
cy
Up
Down
N T
94
1) Overall peak frequency (OPF)
2) Peak frequency of the first 2-second interval of the ECG trace (PFF)
3) Peak frequency of the last 2-second interval of the ECG trace (PFL)
For N vs T classification, we found the maximum value of the OPF for all T signals
in the learning data set to be 6.91 Hz (which is lower than some N cases), whereas the
minimum value of OPF for all N signals in the learning data set was 5.13 Hz (which is
higher than some T cases) (Figure 6.5 a). Therefore, we set 6.91 Hz as the upper limit of
OPF for T cases and 5.13 Hz as the lower limit of OPF for N cases. Among the learning
set, any signal with an OPF N 6.91 Hz was classified as an N signal; and any signal with
OPF b 5.13 Hz was classified as a T signal. For those cases with an OPF between 5.13
and 6.91 Hz, we found that the PFF is greater than the PFL in most T cases. Therefore,
any signal with an OPF between 5.13 and 6.91 Hz was classified as T signal if PFF N
PFL, and N signal otherwise (Figure 6.5 b). Based on these observations, a CART model
with the following decision rules was developed for classifying the N and T cases:
1. If the OPF > 6.91 HZ, then the object is N.
2. If the OPF < 5.13 HZ, then the object is T.
3. For OPF within the interval from 5.13 to 6.91 HZ, the object is classified as T if
the PFF > PFL, and N case otherwise.
This model correctly classified 28 of 30 N and T cases during testing, leading to an
accuracy of 93%.
For the S vs T classification, it may be noted that for every S recording, a
corresponding T recording was taken from the same patient. Therefore, the first step in S
vs T classification involved applying an optimal pairing method to identify the pairs of S
95
and T recordings from the same patient based on the ECG signal morphology information
(Figure 6.6). This information included ECG features such as the arithmetic mean of P-
wave width, height of P wave, QRS width, height of QRS, QT length, T-wave width, and
RR interval from the time domain signal plots.
Figure 6.6 (a) and (b) Learning and testing dataset RR interval and average features
scatter plot, respectively. (Average feature is the average value of mean of QRS, mean of
peak, mean of height of T wave, mean of QT interval, mean of ST segment, and mean of
T duration)
40 60 80 100 120 1400.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
RR interval
Aver
age
Mor
phol
ogy
feat
ure
Learning dataset S and T person grouping
ST
40 50 60 70 80 90 100 110 120 1300.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
RR interval
Aver
age
Mor
phol
ogy
feat
ure
Testing dataset S and T person grouping
ST (b)
(a)
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Table 6.3 S vs T learning data set RR interval and skewness of RR feature table Group S Group T
Person Case RR Interval Skewness RR Case RR Interval Skewness RR a s01 48.28a 1.45 t01 49.62 a 1.37
g s07 96.87 0.50b t07 73.15 1.37 b h s06 110.57 0.42 b t06 110.13 0.85 b i s05 114.80 0.34 b t05 113.77 1.05 b j s03 123.15 0.62 b t03 134.39 1.74 b
b s04 62.97c 1.57 t04 60.65 c 1.02 c s10 66.31 c 0.67 t10 61.08 c 0.84 d s02 71.70 c 0.66 t02 69.33 c 0.72 e s08 74.27 c 0.94 t08 71.58 c 0.89
f s09 92.91 -0.40d t09 88.16 -0.25 d
a Rule 2; b Rule 3; c Rule 4; d Rule 5
Table 6.4 S vs T testing data set RR interval and skewness of RR feature table Group S Group T
Person Case RR Interval Skewness RR Case RR Interval Skewness RR 1 b14 47.47a 1.04 b16 47.58 a 1.64
3 b09 62.96 0.92b b08 71.03 2.51 b
2 b20 53.87c 1.01 b12 51.89 c 1.69 4 b01 64.48 c 0.91 b03 61.93 c 1.26 5 b02 72.72 c 0.84 b07 70.21 c 0.43
6 b04 74.31 0.39d b13 79.92 0.42 d 7 b05 90.01 0.51 d b19 92.74 0.60 d 8 b18 99.53 0.94 d b11 94.50 1.00 d 9 b10 112.67 -0.39 d b15 104.94 -0.38 d
10 b06 119.30 -0.50 d b17 121.35 0.35 d a Rule 2; b Rule 3; c Rule 4; d Rule 5
After grouping ECG traces from the same person, the S and T classification was
performed based on 2 features: average RR interval and skewness of RR interval, which
capture the heart rate and heart rate variability, respectively. Heart rate activities are
97
closely related to the state transitions from soon-to-be-terminating AF to terminating AF.
As shown in Table 6.3 and Table 6.4, the average RR interval was found to decrease
from S to T, whereas the skewness of the RR interval increases from S to T for the same
person, in most cases, from the learning data set. In addition, no S cases were found to
have both higher skewness of RR and smaller average RR interval than T cases for the
same person, except in one case. This finding implies that the RR interval trend in the S
case has either a longer average RR interval or a smaller skewness of the RR interval than
the T case for the same person.
Based on the learning data set, we found that the data can be divided into 2 groups
based on the average RR interval: those whose average RR intervals are greater than or
less than 75 milliseconds. The skewness of the RR interval increases from S to T for
those patients whose RR intervals are greater than 75 milliseconds, and the average RR
interval decreases from S to T for those RR intervals less than 75 milliseconds. However,
if the patient has a very high heart rate, for example, the average RR interval is 50
milliseconds, then it may serve as a major feature instead of the skewness of RR interval
because the higher heartbeat rate will not cause significant RR interval variations unless
there is a sudden disruption in the heart activity. With these findings, both learning data
set (sets S and T) and testing data set (set B) are correctly classified. The CART
classification rules, created based on the learning data for events I and II, are given as
following:
1) Find pairs of recordings (obeject A and object B) from the same patient.
2) If (RR interval < 50) in one pair, then the object with bigger RR interval is T.
(this condition refers to extremely high heart rate (HR=1/RR) case)
98
3) If Skewness RR (object A) 2Skewness RR (object B)
or Skewness RR (object A) 0.5Skewness RR (object B)
� , then object
with greater Skewness for RR is T.
4) If the bigger RR interval for object A and object B <= 75, object A is S if RR
interval decreases from object A to object B.
5) If the bigger RR interval for object A and object B > 75, object A is S, if
Skewness of RR increases from object A to object B.
It may be noted that heart rate analysis based on average RR interval and its variation,
and QRST subtraction techniques using wavelet threshold filter are different from those
reported in the literature.9 In N vs T classification, we used 3 features as inputs for
CART classification, which is one less than the number of features used in Petrutiu et al9,
without significantly compromising on the accuracy (93% for the N vs T testing data set
and 100% for the S vs T testing data set). Furthermore, the CART decision tree has the
potential to integrate medical expertise to generate a number of clinical decision rules for
improving the diagnostic accuracy for different AF states.
6.5 Summary
The CART model patterned after a previous such attempt yielded the best results for
N vs T and S vs T classifications. Interestingly, the CART classifier for N vs. T uses one
less feature than in the previous work [8], and it was found to yield over 95% accuracies
for classifying between N vs. T. Also, application of three decision rules is sufficient to
classify between N vs. T.
CART classification of S vs. T is found to depend on pairing the ECGs of every
individual subject (one taken when the subject is at S condition and the other at T). The
99
optimal pairing algorithm was found to achieve a fairly accurate pairing using the
aggregate ECG features. Based on the pairing information, five decision rules were
generated to derive the CART classifier for S vs. T. From a practical standpoint the
decision rules obtained through CART can be easily coded in the form of a rule-based
diagnostic system, and can be useful for automatic classification between S and T events.
The CART models constructed using temporal relationships among features yielded
high accuracy for both N vs T and S vs T classifications. These results matched the best
results reported in the 2004 Challenge Competition [6] for N vs T classification.
To improve the accuracy of identification of various AF episodes further, it is
necessary (a) to get access to a significantly larger database of both normal and AF cases
and (b) to seek expertise of the cardiologists in providing relevant features and
diagnostics for various cardiologic disorders.
6.6 References
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Anatomically to the Macroreentrant Circuit?," Journal of Electrocardiology, vol.
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complex at different ECG sampling rates," Computers in Cardiology, pp. 495-
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Medical Engineering and Technology, pp. 7-15, 2002.
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[39] R. C. Graeber, "Wavelet packet-based detection of QRS complexes and study of
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pp. 33-36, 2002.
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CHAPTER VII
MYOCARDIAL INFARCTION IDENTIFICATION USING RECURRENCE
QUANTIFICATION ANALYSIS (RQA) OF VCG SIGNALS
7.1 Introduction
According to the U.N. World Health Organization (WHO), myocardial infarction
(MI), also known as coronary heart attack, is a leading cause of fatality in the world. It
results from the occlusion of the coronary artery and insufficient blood supply to
myocardium [1]. It can take place in the inferior, septal, anteroseptal, anterior, apical, or
lateral parts of a heart, and it often leads to necrosis or death of myocardial cells in the
nearby heart locations. Onset of MI can also alter the underlying electrical potential fields
and cardiovascular dynamics. Signals from conventional 12-lead electrocardiograms
(ECG) and/or 3-lead vectorcardiogram (VCG) signals, recorded at the surface of the
body, are widely used in the current methods to detect MI. These signals are generated
due to the sequence of electrical activities taking place in the human heart.
Several methods have been used previously for detecting MI using 12-lead ECG [2,
3]. Also, most of the previous approaches used conventional ECG features, such as
variations in ST wave segments, depth of Q-wave, ST elevation, inversion of T-wave,
and other event intervals [4-6]. Consideration of nonlinear stochastic dynamics
underlying the measured signals, especially with the use of more intuitive VCG has not
been reported in the literature. In this chapter we study the variations in heart dynamics
106
underlying the measured VCG loops obtained from healthy control (HC) vs. Myocardial
Infarction (MI) subjects through the application of recurrence quantification analysis
(RQA) [7], a technique borrowed from nonlinear dynamics systems theory, and MI
detection using neural network (NN) classification of recurrence quantifiers. Recurrence
quantifiers were found to be useful means to detect cardiological disorders, such as atrial
fibrillation (AF) [8]. The present work follows our previous investigations based on using
wavelets to compactly capture information from ECG signals [9] and to use advanced
classification techniques to identify Atrial Fibrillation (AF) episodes [10].
The VCG data [368 MI and 80 HC recordings] available in the PhysioNet Database
[11] was analyzed in this investigation. Each of these recordings contains 15
simultaneously recorded signals, namely, the conventional 12-lead ECG and the 3-lead
Frank (XYZ) VCG. The signals were digitized at 1 kHz sampling rate with a 16-bit
resolution over a range of ±16.384 mV. The 80 HC recordings are acquired from 54
healthy volunteers, and 368 MI recordings are acquired from 148 patients. The recordings
are typically ~ 2 min in duration, and all the signals were recorded for at least 30 sec
long.
The results of recurrence quantification analysis of VCG signals show that
laminarity, determinism, and other properties of VCG trajectories (defined in Appendix
A [12, 13]) fromHC segments are significantly different from those of MI patients. A
radial basis neural network (NN) that uses these recurrence quantities as input features
was able to provide an effective classification (>90% accuracy) between HC and MI. The
remainder of this chapter is organized as follows: Section 7.2 presents the methods used
in the analysis, Section 7.3 contains the overall implementation approach as well as
107
results from feature extraction and MI classification for the PhysioNet PTB database, and
Section 7.4 presents discussion and conclusions arising out of this investigation.
7.2 Research methodology
A human heart is essentially an autonomous electro-mechanical pump. The
vectorcardiogram (VCG) signals capture the heart potentials as a dynamic cardiac vector
in three orthogonal components [14]. VCGs are mutually orthogonal bipolar
measurements obtained from electrodes placed at appropriate locations of the torso as
shown in Figure 2.3.
Figure 7.1 Coordinate system used for obtaining VCG signals
VCG are not as commonly used in the medical practice as the 12-lead ECG
because, among other reasons, the interpretations of 3D VCG loops are not commonly
108
taught and require specialized knowledge of spatial decomposition of the heart vector.
Since X, Y, and Z constitute the three axes, and the time, the fourth axis, they cannot be
displayed on a paper. For this, dynamic vectorcardiography is needed, which can be
displayed only on a computer screen. Some ECG monitoring equipment uses the VCG
for calculation and storage purposes, but present the derived 12-lead ECG for the
interpretation by medical doctors. However, for automated computer analysis and
diagnosis both 12-lead ECG and 3-lead VCG are equally applicable.
Figure 7.2 A representative VCG plot showing the vector loops for P, QRS, and T wave
activities
Dower and his group [15-17] have conducted pioneering studies on VCG signal
analysis and demonstrated that the 3-lead VCG and 12-lead ECG can be linearly
transformed from one to the other without significant loss of clinical information
regarding heart dynamics. Also, since VCG requires fewer leads to be placed on the
-0.50
0.51 -0.2 0 0.2 0.4 0.6 0.8
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Vz
Vx
Vy
P loop
T loop
QRS loop
109
subject, it offers additional advantages for remote and continuous patient monitoring.
Therefore, it is imperative to analyze the VCG signals and explore the relationships with
cardiovascular conditions.
The VCG vector loops contain 3D recurring, near-periodic patterns of heart
dynamics. Each heart beat consists of three loops corresponding to P, QRS, and T wave
activities. The 3D loops can be visualized in the X-, Y- and Z- space domain with time
entering implicitly (see Figure 7.2). Identification and extraction of a compact set of
features from these patterns is critical for effective detection and diagnosis of various
ailments. Complex nonlinear stochastic dynamics (of human cardiovascular system)
underlying these VCG vector loops pose significant challenges to the extraction of
features that can track cardiovascular anomalies. Some previous studies have
characterized this nonlinear stochastic dynamics of human heart to exhibit low
dimensional chaos [18-20].
Features extracted based on conventional statistical approaches and linear system
perspectives alone tend to have limitations for capturing signal variations resulting from
changes in the cardiovascular system dynamics resulting from different myocardial
damage levels. The use of a rigorous nonlinear analysis based on combining multifractal
embedding principles [21] with recurrence analysis [22, 23] can lead to extraction of
effective features. The strength of these methods emerges from their effective use of the
recurring patterns of loops exhibited by VCG and many other physiological signals.
Through effective use of recurrence properties, it is possible to specify an appropriate
tight set of features that track the relevant cardiovascular state variations and are
insensitive to other extraneous variations.
110
Figure 7.3 A graphical illustration of relationship among a VCG, the X,Y,Z time series
and the unthresholded recurrence plot
The recurring patterns in the nonlinear system state space can be captured using
recurrence plots. It provides a convenient means to capture the topological relationships
existing in the 3D VCG vector space in the form of 2D images. As shown in Figure 7.3,
the unthreshold recurrence plot (UTRP) delineates the distances of every point x(i),
realized at time index i in a VCG, to all the others, i.e., ( , ) : ( ( ) ( ) ) �D i j x i x j0 , where
||·|| is a distance measurement (e.g., the Euclidean norm) and �(·) is the color code that
maps the distance to a color scale. As shown in Figure 7.3, the color code of the distance
between the ith and jth embedded vectors is located in coordinate (i,j) of the recurrence
plot. If the color code at the point is blue, then the points are located close to each other
111
in the VCG, and if the color code is red, the points are located farther apart in a VCG.
While the thresholded recurrence plot (TRP) methods (see Figure 7.4) only draw points
when the distance ( ) ( )x i x j� between two vectors is below a cutoff distance r:
( , ) : ( ( ) ( ) )T i j r x i x j 1 � � , where � is the Heaviside function [22, 23].
Figure 7.4 A representative thresholded recurrence plot of VCG
Thus, a recurrence plot shows the times at which a state of the dynamical system
exhibits recurrence, i.e., the time-pairs at which the trajectories of a system evolution
come within a specified neighborhood. The structures of a recurrence plot have distinct
topology and texture patterns (see Figure 7.3 and Figure 7.4). The recurring dark (blue)
diagonal (45o) lines indicate the near-periodicity of the system behaviors over given time
segments with a period, heart rate, equal to the separations between successive diagonal
lines. Recurrence-based methods have recently shown potential for representation, de-
noising and prediction of measurements from complex systems. This present work
112
involves extracting representative nonlinear dynamical features from the recurrence plots
of VCG vector loops for classifying between MI and HC. The classification techniques
used are based on neural networks (NN) and linear discriminant analysis methods which
can lead to effective mapping of the features that capture the symptoms of MI conditions.
7.3 Implementation and results
Figure 7.5 Flow diagram showing research methodology used
The overall approach implemented for analyzing the nonlinear dynamics underlying
VCG signals and classifying between HC and MI are summarized in Figure 7.5 and
detailed in the following subsections, consists of the following three steps:
VCG Analyzer
Feature Extraction
Recurrence Quantification Analysis (RQA)
Classification
Linear Discriminant Analysis
Neural Network
VCG Signal
Recurring pattern features
Feature Analysis
PCA Transformation
Histogram Comparison
Transformed features
113
(1) Recurrence quantification analysis (RQA): Presentation of the 3D VCG signals in
the form of a 2D unthresholded recurrence plot so that mathematical description
of the salient patterns contained in the signal as well as the procedures for feature
extraction can be much compactly captured, and nonlinear dynamic quantifiers
that describe the specific recurring patterns in the VCG are efficiently extracted.
(2) Feature analysis and conditioning: Orthogonalizing the recurrence quantifiers so
that the significant correlations that exist between some of the recurrence
quantifiers are eliminated, leading to the removal (projecting out) of the redundant
information in the feature set. The extracted features should be sensitive to the
state variables to be estimated.
(3) MI classification: Associating, through both linear and nonlinear maps, the
extracted features with an appropriate representation of unknown state variables.
7.3.1 Recurrence quantification and feature extraction
Recurrence plots are graphical displays of signal patterns, but quantitative features
need to be extracted to analyze the underlying processes and detect hidden rhythms in the
graphics. Statistical quantifiers of complexity, such as recurrence rate (�), determinism
(DET), linemax (LMAX), entropy (ENT), laminarity (LAM), and trapping time (TT)
serve as such typical recurrence quantifiers that can be estimated from an appropriately
constructed Threshold Recurrence Plots (TRP) [12]. The definitions of the above six
recurrence quantifiers and their relationships with heart dynamics are summarized in
Appendix A [12, 13]. These quantitative features are extracted based upon the patterns
across the diagonal or vertical lines in the thresholded recurrence plot.
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Figure 7.6 Histogram comparisons between HC (filled green) and MI(not shaded) for the six RQA quantifiers, namely, recurrence rate (�), determinism (DET), linemax (LMAX), entropy (ENT), laminarity (LAM), and
trapping time (TT) (Histogram x-axis is feature value, and y-axis is the normalized frequency)
0 2 4 6 8 10 120
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Figure 7.6 shows the histogram comparisons between HC and MI for the six RQA
quantifiers �, DET, LMAX, ENT, LAM, and TT extracted from VCG signals. It may be
noted that quantifier ranges of the six recurrences for HC are higher than those for MI,
implying that dynamic properties of HC are more volatile than MI. The more complex
variability of heart dynamics shown in the HC cases are also pointed out in Goldberger’s
study on fractal dynamics in physiology [20]. The distinct distributions across the MI and
HC groups provide effective raw features for the following principal component analysis
(PCA) and neural network (NN) classification.
7.3.2 Feature Analysis
The six recurrence features were further tightened by projecting out redundant
information, and made more sensitive to the relevant variations in underlying dynamics.
Principal component analysis (PCA) techniques are commonly used to reduce the
dimension of the input (feature) vectors in situations where the dimension of the input
vector is large but the components of the vectors are highly correlated (redundant). PCA
orthogonalizes the components of the input vectors (so that they are uncorrelated with
each other), and orders the resulting orthogonal components (principal components) so
that those with the largest variation are considered first, and projects those components
that contribute the least to the variation in the data set [9].
PCA transforms the original six nonlinear dynamical features into six orthogonal
principal components, named F1, F2, …, F6. The first principal component captures the
largest variations in the nonlinear dynamical feature datasets, and the principal
components are uncorrelated thus facilitating linear classification. We have found from
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our earlier investigations that linear classifier is more accurate for uncorrelated principal
components than the correlated original nonlinear dynamical features [10].
Figure 7.7 shows the histogram distribution of PCA transformed feature values for
MI (bars not shaded) and HC (shaded green). Histograms of F1 show a statistically
significant difference between the distributions for MI vs. HC, as gathered from a two-
sample Kolmogorov-Smirnov test. Thus, the first principal feature F1 captures most of
the differences between MI and HC. Histogram distributions for the other features,
especially F3, F4, and F5 also show some differences, but not as statistically significant
as F1, between MI and HC. These features can provide additional degrees of freedom,
especially for classification with nonlinear discriminant boundaries.
Table 7.1 Linear model misclassification results using F1 and other principal features (F2 F6)
Features Misclassification resultsMI (368) HC(80)
F1 and F2 12 34F1 and F3 17 30F1 and F4 19 32F1 and F5 14 33F1 and F6 20 31
This is further evident from feature distribution plots shown in Figure 7.8 (a) to (e).
They show the plots of the principal feature F1 against the other 5 principal features (F2
through F6). A genetic algorithm is used to determine the optimal linear boundary in the
feature space separate the feature values of MI (red +) from the HC (blue �) feature
values. This boundary line maximizes the discrimination, and hence the classification
accuracies using every pair of features formed by F1 and other five principal features
(F2-F6). The clear separation of HC and MI across F1 axis, marked by < 45º deviation
from the vertical of the linear boundary indicates that F1 is the most significant
discriminating feature.
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Figure 7.7 Histogram distributions of PCA feature values F1 to F6 for MI (not shaded) and HC (shaded green) (Histogram x-axis is feature value and y-axis is the normalized frequency)
-100 -50 0 50 100 150 2000
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Figure 7.8 (a) to (e) Distribution of values of features F2 through F6 vs. F1 for MI (red �) and HC (blue +) subjects with linear separator obtained using a genetic algorithm
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Table 7.1 shows the linear model misclassification results using F1 and other
principal features (F2 through F6). As will be demonstrated in Section III.C, the linear
classification with all six features can distinguish ~ 97% of MI patients. The features
extracted from recurrence analysis are, therefore, adequately discriminatory to capture the
dynamic variations between HC and MI, and ensure a high classification accuracy for the
MI patients. It may be noted that a non-linear model can further increase the
classification accuracies.
Figure 7.9 Relative importance of RQA quantifiers in the first principal feature (F1)
Since F1 is clearly the most discriminating feature, we next examine the relative
contributions of the six RQA features for F1. As shown in the Figure 7.9, LAM, DET and
LMAX are the most important features capturing the corresponding pathological
variations of myocardial infarction in the first principal feature F1. PCA results show that
MI conditions are related to stability, mixing rates of VCG trajectories, and periodicity in
the cardiovascular system.
DET LMAX ENT LAM TT0
0.2
0.4
0.6
0.8
Features
Dire
ctio
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osin
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7.3.3 Classification
Linear discriminant analysis was used first to facilitate training of a simple
classifier to capture the complex relationships connecting the PCA transformed features
and MI. A classifier essentially is a model that maps the features to the patient’s health
conditions, namely, healthy control (HC, coded -1) or myocardial infarction (MI, coded
+1). The states are coded such that a mathematical mapping between the feature values
and the patient’s conditions is possible. During the model building or the training phase
of the classifier, the model parameters are adjusted so that, when features extracted from
VCG signals of HC cases are presented, model outputs a value closer to -1, and so on.
Linear discriminant analysis assumes that a linear hyperplane serves as a boundary
separating HC and MI points in the feature space, and it uses a least square fit to
determine the coefficients which give the best fit. The hyperplane that captures the
relationships between the input features and the output response is of the form
0 1 1 2 2 ... n ny a a x a x a x #
where ai are the coefficients and xi represents the input features. The residual error � of
the difference between the actual value and the model output determines whether the
presented VCG is taken from HC or MI subject. As shown in Table 7.2, the magnitude
of ia* 2 coefficients as well as the corresponding p-values1 are calculated to test the
statistically significant levels of the features used. Larger the ia* 2 value, the more
important is the corresponding feature. Contrarily, smaller the p-value, the more
significant is the feature. Table 7.2 shows that the contributions of the principal features
1 The p-value is that probability that any variation in the input feature will result in a statistically significant change in the output response, and p-value should be < 0.05 for the input feature to be significantly discriminating.
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F1 followed by F3, F4, and F5 which are the most discriminating features for MI vs. HC.
These results are consistent with the initial observation on the discriminatory power of
the principal features.
Table 7.2 Relative importance of principal features F1 to F6 using linear regression Feature F1 F2 F3 F4 F5 F6
PCA eigenvalue 2237.1857 205.1453 15.0914 2.0983 0.5264 0.0503* 47.2989 14.3229 3.8848 1.4486 0.7255 0.2243
Linear regression coeff. ai -0.0098 0.0017 -0.0314 -0.0469 0.1382 0.2908| ia* 2 | 0.4561 0.0582 0.1505 0.0841 0.1341 0.0241 p-value 0 0.076 0 0.0129 0 0.448
Table 7.3 Results of linear regression classification for MI and HC
S. No. Training data points % Testing accuracy*
MI (368) HC (80) MI HC Mean Std Mean Std
1 332(90%) 72(90%) 97.75 2.46 53.50 14.27 2 295(80%) 64(80%) 97.42 1.87 54.44 10.82 3 276(75%) 60(75%) 96.77 1.99 56.15 10.10 4 246(67%) 54(67%) 96.98 1.61 54.85 7.63 5 184(50%) 40(50%) 97.22 1.37 53.40 6.85 6 123(33%) 27(33%) 96.43 1.53 53.21 7.15 7 74(20%) 16(20%) 96.19 1.90 52.22 9.35 8 37(10%) 8(10%) 94.60 3.48 49.99 13.08
* accuracy is only calculated for testing dataset
As shown in Table 7.3, the linear classifier can identify MI very accurately (~ 97%
accuracy) with high consistency (small standard deviations). Such high classification
accuracies indicate that the features extracted from the recurrence analysis are significant
discriminating features even for the linear classifiers. However, it can be seen that the
accuracy of the linear classifier for HC is rather low. This is likely because, as will be
shown later, the current HC samples are significantly sparse to capture the diverse
variations in cardiac dynamics of HC subjects. The HC subjects show more variations in
the heart dynamics and poses difficulties for the linear classifiers.
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Next, a radial basis function (RBF) neural network (NN) was used to classify HC
and MI cases [24]. An advantage of using NN is that it can provide a nonlinear boundary
separating MI and HC cases. In particular, the RBF network uses radial basis function,
which can take advantage of the clustering and separation of the feature values in the
feature space to reduce the amount of training as well as the need for extensive input
patterns. The functional form relating the features to the heart condition can be different
at different locations of the feature space. This is particularly useful because the
distribution of features in 2-D plots in Figure 7.7 show a considerable grouping of the
feature values of HC and MI groups.
Figure 7.10 Radial basis function (RBF) neural network (NN)[24]
Figure 7.10 is a schematic of the radial basis function (RBF) neural network (NN)
used in this investigation. The network consists of 2 layers – a radial basis layer and a
linear layer. In the first layer, distances (difference) of the input feature vector p1 from
rows of the weight matrix W1 are computed. It is then multiplied by the bias b1 to form
net input n1 going into the neuron. Each row of the weight matrix W1 acts as a center for
a transfer function. The transfer function should be local, i.e., the response of the transfer
function should be maximum near the center and should vanish to zero as the inputs
disperse away from the center. The activation a1 of layer 1 is obtained by transforming n1
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using a symmetric transfer function. Most common choice for the transfer function is
Gaussian function. The Gaussian function responds only to a small region of the input
space where the Gaussian is centered. The second layer of the network is a linear layer.
Training RBF network involves adjusting center and spread of the Gaussian function.
The results of RBF classification are summarized in Table 7.4. It shows that the
RBF neural network improved the classification accuracy of both MI and HC (see Table
7.3 and Table 7.4) compared to linear classifier by using the nonlinear classification
boundary and local analysis. It may also be noted that HC subject conditions are more
diverse than the MI subject. Notably the RBF classifiers improved the accuracy of HC by
20% to 75%.
Table 7.4 Results of neural network (NN) classification for MI and HC
Training data points Number of
neurons
% Accuracy for MI classification
% Accuracy for HC classification
Training Testing Training Testing MI (368) HC (80) Mean Std Mean Std Mean Std Mean Std
332 (90%) 72 (90%) 350 99.60 1.24 98.70 1.98 78.60 5.63 75.00 6.14 295 (80%) 64 (80%) 250 99.10 1.63 98.40 2.12 71.60 6.18 71.50 7.13 220 (60%) 72 (90%) 200 98.82 2.59 96.62 2.86 73.20 5.81 75.00 6.23 220 (60%) 64 (80%) 200 96.84 2.61 96.40 2.94 69.10 6.28 68.20 6.89 148 (40 %) 48 (60%) 150 95.74 5.96 94.81 6.89 65.80 10.86 65.62 12.21 148 (40%) 32 (40%) 150 95.48 6.14 92.27 6.58 58.93 12.28 56.25 14.45
Total Number ECG data sets: MI 368 and HC 80 data sets
We also investigated the influence of the number of training and testing patterns on
the accuracy of the classifier. Figure 7.11 shows the variation of the percentage accuracy
with the number of points used in RBF training. We chose the same number of data
points for MI as HC subjects. It can be seen that the accuracy of MI cases has come down
and show a similar trend for both HC and MI when the number of training data points is
sparse (<100), implying that availability of more HC data can improve the accuracy
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further. Also, since the accuracy of MI subjects improves very slowly at higher
accuracies (>90%), significant data need to be available if we were to increase the
accuracy to two 9’s (99%), or three 9’s (99.9%). Also, from examining the variation of
classification accuracies for MI and HC with the sizes of training and testing patterns as
shown in Figure 7.11, it is evident that the accuracies would improve for HC with
increase in the number of training data points.
Figure 7.11 Variation of % accuracy of the neural network (NN) (both training and testing) with the number of data points for MI and HC
7.4 Summary
In this chapter, an attempt was made to analyze the VCG signals using nonlinear
dynamics approaches by applying the recurrence quantification analysis (RQA) for
feature extraction to distinguish MI patients from the HC subjects. A linear classifier was
found to correctly identify MI about 94% of the times. However the linear classifier was
able to correctly classify HC cases about 45% of the times. A neural network classifier
was used to improve MI classification accuracies to about 97%, and HC to about 75%. It
was also shown that the classification accuracies for HC cases are likely to improve with
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the availability of additional VCG data. These results indicate that the VCG features
extracted using RQA are good indicators of the cardiovascular conditions, and the NN
classifiers capture the nonlinear relationships connecting the recurrence features with
patients’ conditions. Based on these findings, the following conclusions may be arrived.
1. The recurrence plot representation of VCG signals captures the complex patterns
buried in the VCG vector loops and the underlying nonlinear dynamics. The study of the
patterns quantified using the RQA were found to provide useful information for detecting
MI.
2. It was found that many of the VCG features extracted using RQA contained highly
redundant information. Principal component analysis (PCA) is used to reorient the
features to align in such a way as to remove significant correlation between features. The
principal feature F1 was found to be the most discriminatory feature and the VCG
features LMAX, LAM, and DET extracted using RQA are the three leading contributors
for F1.
3. Results of linear discriminant analysis showed strong relationships between the
extracted recurring patterns related features and the patients’ cardiovascular conditions.
Over 90% of the MI signals were correctly classified using the principal features.
However, the linear classifier was not adequate to identify HC subjects (~50%). This was
somewhat surprising as one would expect this to be the case for MI patients (due to
various types of MI) than HC subjects. This is likely a limitation of the linear classifier in
capturing the diverse patterns of HC subjects.
4. Neural network model identifies the nonlinear mapping from the extracted features
to the patients’ conditions and provides high accuracies for MI classification (>95%),
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which make the real-time and practical implementation plausible. It also improves the
accuracy for identifying HC to above 75%. It was also shown that the training and testing
accuracies can improve with the availability of more VCG recordings, especially of the
HC subjects, for training and testing.
5. Investigation of the effects of the number of training and testing patterns on the
percent accuracy of the classifier has found similar trends for both HC and MI, implying
that availability of more data would improve the accuracy further in the case of HC. Also,
from examining the variation of classification accuracies for MI and HC with the sizes of
training and testing patterns, it is evident that the accuracies will be improved with an
increase in the size of training data points.
6. It appears that improve the accuracy of identification of HC subjects as well as
various myocardial infarction (MI) patients to double nines (99%) or triple nines (99.9%),
the following are needed: (1) access to larger data collections of myocardial infarction
and health control cases, and (2) expertise of the cardiologists in providing relevant input
parameters and analysis of the data.
7.5 References
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[2] M. C. MacLachlan, B. F. Nielsen, M. Lysaker, and A. Tveito, "Computing the
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2006.
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[3] P. Porela, H. Helenius, and L. M. Voipio-Pulkki, "Automated ECG injury scores
in the prediction of the myocardial infarction (AMI) size," in Computers in
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[4] D. Lemire, C. Pharand, J. Rajaonah, B. A. Dube, and A. R. LeBlanc, "Wavelet
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[6] K. K. Talwar, S. Radhakrishnan, V. Hariharan, and M. L. Bhatia, "Spatial
vectorcardiogram in acute inferior wall myocardial infarction: its utility in
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the analysis of complex systems," Physics Reports, vol. 438, pp. 237-329, 2007.
[8] R. Sun and Y. Wang, "Predicting termination of atrial fibrillation based on the
structure and quantification of the recurrence plot," Medical Engineering &
Physics, vol. In Press, Corrected Proof.
[9] H. Yang, S. T. Bukkapatnam, and R. Komanduri, "Nonlinear adaptive wavelet
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[10] S. Bukkapatnam, R. Komanduri, H. Yang, P. Rao, W.-C. Lih, M. Malshe, L. M.
Raff, B. Benjamin, and M. Rockley, "Classification of atrial fibrillation episodes
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components of a new research resource for complex physiologic signals,"
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"Recurrence plot based measures of complexity and their application to heart-
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[14] J. Malmivuo and R. Plonsey, Bioelectromagnetism: principles and applications of
bioelectric and biomagnetic fields. USA: Oxford University Press, July 18, 1995.
[15] G. E. Dower and H. B. Machado, "XYZ data interpreted by a 12-lead computer
program using the derived electrocardiogram," Journal of electrocardiology, vol.
12, pp. 249-261, 1979.
[16] G. E. Dower, H. B. Machado, and J. A. Osborne, "On deriving the
electrocardiogram from vectorcardiographic leads," vol. 3, p. 87, 1980.
[17] G. E. Dower, A. Yakush, S. B. Nazzal, R. V. Jutzy, and C. E. Ruiz, "Deriving the
12-lead electrocardiogram from four (EASI) electrodes," Journal of
electrocardiology, vol. 21, pp. S182-S187, 1988.
[18] P. C. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z.
R. Struzik, and H. E. Stanley, "Multifractality in human heartbeat dynamics,"
Nature, vol. 399, pp. 461-465, 1999.
[19] H. E. Stanley, "Statistical physics and physiology: monofractal and multifractal
approaches," Physica A: Statistical Mechanics and its Applications, vol. 270, p.
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[20] A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. C. Ivanov, C. K. Peng,
and H. E. Stanley, "Fractal dynamics in physiology: alterations with disease and
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aging," Proceedings of the National Academy of Sciences, vol. 99, pp. 2466-2472,
February 19, 2002 2002.
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CHAPTER VIII
VECTORCARDIOGRAPHIC OCTANT FEATURES FOR THE DIAGNOSTICS
OF HETEROGENEOUS MYOCARDIAL INFARCTION
8.1 Introduction
Coronary heart disease is the leading cause of death in the United States (America
Heart Association - AHA). Myocardial infarction, also known as heart attack, is resulted
from the occlusion of the coronary artery and insufficient blood supply to myocardium. It
can take place in the anterior, inferior, posterior, infero-lateral, anterior-septal, posterior-
lateral etc. parts of a heart. The myocardial infarction triad is ischemia, injury, and
necrosis, and any of the three may occur alone. Ischemia is normally from the reduced
blood supply, injury indicates acuteness of infarct, and infarction is the symptom of
myocardium necrosis [1].
The 12-lead electrocardiograms (ECG), recorded at the surface of the body, are
widely used in clinical diagnostic applications of myocardial infarction. The ECG
diagnosable myocardial infarctions can be divided as following:
� Q wave infarction, which is diagnosed by the presence of pathological Q waves.
A pathological Q wave is defined as at least 40 ms duration or 1/3 downward
deflection of the entire QRS magnitude, and found in most 12-lead ECG except
lead III and lead aVR.
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� Non-Q wave infarction, which is diagnosed in the presence of ST elevation or
depression, and T wave abnormalities.
It is conceptually known that 12-lead ECG examines the same heart activity from
12 different perspectives of view, and vectorcardiogram (VCG) is only 3-lead orthogonal
heart electrical activity measurements. The VCG vector loops contain 3D recurring, near-
periodic P, QRS, and T wave activity patterns in eight octants. Dower et al.[2-4] reveals
that 12-lead ECGs can be linearly transformed to 3-lead VCGs without a significant loss
of clinically useful information regarding heart dynamics. But VCG trajectory is not as
commonly utilized as 12-lead ECG traces because of spatial imagination requirements
(for e.g., rotation and projection of VCG trajectory), the loss of temporal information, and
medical doctors’ accustomed trainings of 12-lead ECG interpretation for one hundred
years. However, VCG will be a better option for computer processing and analysis
because it overcomes not only the information loss from only one or two ECG signals but
also the dimensionality problems induced by the 12-lead ECG.
The VCG recordings (368 MIs and 80 HCs) available in the PhysioNet PTB
Database[5, 6] were considered in this study. Each of these recordings contains 15
simultaneously recorded signals, namely, the conventional 12-lead ECG and the 3-lead
Frank (XYZ) VCG. The signals were digitized at 1 kHz sampling rate with a 16-bit
resolution over a range of ±16.384 mV. The 80 HC recordings are acquired from 54
healthy volunteers, and 368 MI recordings are acquired from 148 patients. Within the MI
recordings, there are 89 inferior, 56 infero-lateral, 19 infero-postero-lateral, 1 infero-
posterior, 47 anterior, 43 antero-lateral, 77 antero-septal, 2 antero-septo-lateral, 3 lateral,
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4 posterior, 5 postero-lateral and 22 unknown site cases. Those VCG recordings are
typically ~ 2 min in duration, and all the signals were recorded for at least 30 sec long.
8.2 Feature extraction
To characterize and quantify heterogeneous myocardial infarction, three groups of
features are collected as following:
(1) Conventional ECG features: mean and standard deviation of RR interval, QT
interval, ST elevation or depression;
(2) VCG vector features: magnitude of Q wave, R wave, T wave vector, angle
between R and T vector, azimuth R vector angle, and azimuth T vector angle;
(3) VCG octant features: octant location of Q, R, T vectors, percentage of vector
points in an octant, and maximum, average, variance of vector magnitude
distributions in each octant.
Figure 8.1 Frank XYZ VCG ensembles, (a) X-direction; (b) Y-direction, (c) Z-direction
Traditional ECG intervals, for e.g., RR and QT intervals, are highly variable
measurements. The heart rate variability, respiration, artifacts, power-line interference are
known to generate ECG near-periodicities and influence the interval calculation. As
shown in Figure 8.1, we calculate the mean and standard deviation of those intervals
(a) (b) (c)
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through the extracted heart beat ensembles over time. Figure 8.1 (a) (b) (c) illustrate the
Frank X, Y, Z ensembles. It may be noted that same direction ensembles are sharing
similar morphologies, but there are still beat-to-beat variations due to heart rate
variability. The group of conventional interval features is extracted as RR average (RR
avg), RR standard deviation (RR std), QT average (QT avg), QT standard deviation (QT
std). The ST segment integration area (ST_itg) is determined by computing the areas
from J point (location where QRS complex joins the ST segment) to J+80 ms if heart rate
(HR) is less than 100 beat/min (or J+72 ms if 100<HR<110, or J+64 ms if 110<HR<120,
or J+60 ms if HR>120) [7]. Since myocardial infarction often causes the ST elevation or
depression in 12-lead ECG signals, the drifts from isoelectric line, therefore, will also
result in the increase of VCG ST segment area.
Figure 8.2 VCG attractor and its 2D projection plots with Q, R, T vectors
Q
R
T
R
TQ
Q
T
R
R
TQ
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The group of VCG vector features are designed to capture the myocardial infarction
pathological patterns, for e.g., Q vector length (Q_mag) will help identify the Q wave
infarction. Increasing of T wave amplitude is related to the T vector magnitude (T_mag).
R vector magnitude (R_mag) indicates the size of the ventricular chambers and proximity
of chest electrodes to ventricular chamber etc. As shown in Figure 8.2, the
aforementioned Q, R, T vectors are from the isoelectric point in VCG to the remotest
points in the respective Q, R, T loops. In addition, the angle between R and T vector are
computed to provide more information about the myocardial ischemia, injury, necrosis
and their locations in myocardium. Previous study shows that QRS loop and T loop are
close to each other in normal cases, while far from each other in some cardiovascular
diseases [8].
(a) (b)
Figure 8.3 (a) VCG electrodes placement in human body; (b) the corresponding
octant definitions
As shown in Figure 8.3, a new octant numbering system is designed to be used with
the VCG measurement system. Binary code 0 is employed to represent the negative
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direction in X, Y, or Z axis, and 1 is for the positive direction. For instance, if the octant
lays in all negative XYZ directions, binary coding will be 000 and the resulted octant
number is also 0. The details of eight octant binary coding are shown in Table 8.1. Based
upon the VCG electrode placements in the human torso, the coordinate system in Figure
8.3 (b) is rotated to illustrate the exact position of every octant in the human body. All the
octants with even numbering are found to be in the anterior locations, and odd in the
posterior. Both octant 0 and octant 2 are found to be located in the right anterior side.
This binary octant definition system does not need to be memorized but deduced, and is
easier for users to find the octant spatial location.
Table 8.1 VCG octant definition and color mapping
Octant X Y Z Binary code Location (X, Y, Z) Color code 0 �3 �3 �3 (000) Right -superior-anterior Black 1 �3 �3 3 (001) Right -superior-posterior Blue 2 �3 3 �3 (010) Right -inferior-anterior Gray 3 �3 3 3 (011) Right -inferior-posterior Cyan 4 3 �3 �3 (100) left-anterior-superior Magenta5 3 �3 3 (101) left -superior-posterior Green6 3 3 �3 (110) left -inferior-anterior Yellow7 3 3 3 (111) left -inferior-posterior Red
It may be noted that former VCG studies about the diagnostics of cardiovascular
diseases (CVD) investigated T loop morphology[8-10], angles of QRS, T loops, QRS
loop area[11-15], VCG attractor fractal dynamics[16], QRS and T vector length [17-19]
etc., but few approaches investigated the relationships of CVDs to cardiac vector length
distributions in each octant and T vector octant locations. Our dynamic VCG
representation study in chapter V gave some clues about the intrinsic relationships
between cardiovascular diseases and VCG octant information. As illustrated in Figure
8.4, the healthy control has significant differences in the VCG octant distributions from
136
myocardial infarction. Eight specific colors are used to paint the vector points in each
octant (see Table 8.1). For instance, it may be noted in Figure 8.4 that red line segments
(octant 7) have different magnitudes and percentages to the whole trajectory for HC and
MI. Therefore, the group of VCG octant features is investigated for a high separation
probability of myocardial infarction from healthy control. T vector position will locate
those T wave inversion and abnormal cases. R vector octant provides information about
the heart electrical axis. Vector point magnitude and proportion in all eight octants
indicates the cardiac vector changing directions and strengths in different local areas etc.
Figure 8.4 VCG trajectories with different octant colors in the 3D coordinate system, (a) HC; (b) MI
137
8.3 Feature analysis
The Kolmogorov–Smirnov goodness-of-fit hypothesis test (KS test) is performed to
examine the differences in both location and shape of empirical cumulative distribution
functions of the forty six features between HC and MI samples. The feature sample
empirical distribution function En for n iid observations Xi is defined as
1
1( ) ( )n
n ii
E x I X xn
�� ,where ( )iI X x� is the indicator function. The Kolmogorov-
Smirnov statistic is _ _sup | ( ) ( ) |n MI n HCx
D E x E x � , which is the maximum difference
between HC and MI feature empirical distribution functions, and the asymptotic p-value
is calculated as 2 20.11exp( 2 ( 0.12 ) )p n Dn
� 2 2��
, where MI HC
MI HC
n nnn n
2
� . If the
probability value p < 0.05, the distribution differences are considered statistically
significant and the hypothesis that the distributions are distinct will be TRUE. Table 8.2
shows the KS test statistics, and the mean and standard deviation of 46 features of the MI
and HC recordings from PTB database.
Figure 8.5 Kolmogorov-Smirnov statistic variations of 46 features
Octant0
Octant2T Wave
138
Table 8.2 Statistical analysis (Kolmogorov-Smirnov test) of three groups of featuresGroup No. Feature Hypothesis p-value KSSTAT(D) MI(mean) MI(std) HC(mean) HC(std)
I
1 RR avg TRUE 7.27E-07 33.0% 811.47 168.84 911.92 152.12 2 RR std TRUE 3.00E-10 40.8% 59.74 262.20 49.33 28.40 3 QT avg FALSE 1.80E-01 13.3% 387.69 46.91 385.32 32.06 4 QT std TRUE 1.28E-02 19.3% 14.37 34.93 12.23 20.50 5 ST_Itg FALSE 3.23E-01 11.6% 1.54 1.00 1.61 0.74
II
6 Q_mag TRUE 5.69E-07 33.3% 0.33 0.19 0.45 0.16 7 T_mag TRUE 2.70E-20 58.0% 0.25 0.14 0.40 0.14 8 R_mag TRUE 2.09E-11 43.2% 1.17 0.39 1.57 0.50 9 Tang TRUE 7.54E-03 20.3% 116.90 31.50 119.35 15.44
10 Rang FALSE 9.63E-02 14.9% 65.66 25.86 68.90 23.60 11 RTang TRUE 3.95E-13 46.4% 101.52 44.88 56.64 28.39
III
12 R_pos TRUE 3.38E-02 17.3% 6.01 1.41 6.46 1.02 13 T_pos TRUE 8.71E-23 61.6% 3.46 2.26 5.88 0.72 14 Q_pos FALSE 3.12E-01 11.7% 0.27 0.92 0.00 0.00 15 Oct0AvgN TRUE 1.48E-27 67.8% 0.09 0.04 0.12 0.01 16 Oct0MaxN TRUE 9.63E-29 69.3% 0.38 0.16 0.54 0.04 17 Oct0Ratio FALSE 6.16E-01 9.2% 0.22 0.16 0.21 0.15 18 Oct0VarN TRUE 4.30E-25 64.7% 0.01 0.01 0.01 0.00 19 Oct1AvgN TRUE 1.24E-09 39.5% 0.10 0.07 0.12 0.02 20 Oct1MaxN TRUE 1.59E-05 29.4% 0.61 0.22 0.74 0.25 21 Oct1Ratio TRUE 9.56E-13 45.7% 0.21 0.16 0.39 0.17 22 Oct1VarN TRUE 1.60E-04 26.4% 0.01 0.01 0.01 0.01 23 Oct2AvgN TRUE 7.26E-23 61.7% 0.11 0.05 0.15 0.02 24 Oct2MaxN TRUE 9.99E-19 55.7% 0.44 0.26 0.62 0.30 25 Oct2Ratio TRUE 4.41E-08 36.0% 0.14 0.11 0.06 0.05 26 Oct2VarN TRUE 1.83E-21 59.7% 0.01 0.01 0.02 0.01 27 Oct3AvgN FALSE 4.57E-01 10.4% 0.09 0.03 0.09 0.02 28 Oct3MaxN TRUE 4.88E-14 48.0% 0.61 0.30 0.82 0.36 29 Oct3Ratio TRUE 2.28E-03 22.3% 0.11 0.10 0.06 0.05 30 Oct3VarN TRUE 5.50E-06 30.7% 0.01 0.01 0.01 0.01 31 Oct4AvgN TRUE 5.50E-06 30.7% 0.10 0.03 0.09 0.01 32 Oct4MaxN TRUE 3.69E-11 42.7% 0.72 0.23 0.64 0.32 33 Oct4Ratio TRUE 1.17E-05 29.8% 0.09 0.08 0.04 0.04 34 Oct4VarN FALSE 2.71E-01 12.1% 0.01 0.01 0.01 0.01 35 Oct5AvgN TRUE 1.12E-05 29.8% 0.17 0.10 0.15 0.09 36 Oct5MaxN TRUE 1.58E-08 37.1% 0.98 0.38 1.22 0.53 37 Oct5Ratio TRUE 1.48E-09 39.3% 0.04 0.05 0.03 0.09 38 Oct5VarN TRUE 3.21E-02 17.4% 0.05 0.07 0.05 0.06 39 Oct6AvgN TRUE 1.84E-11 43.3% 0.13 0.03 0.16 0.03 40 Oct6MaxN TRUE 1.46E-05 29.5% 1.02 0.42 1.30 0.45 41 Oct6Ratio TRUE 1.31E-13 47.3% 0.13 0.10 0.18 0.05 42 Oct6VarN TRUE 5.74E-08 35.8% 0.03 0.02 0.03 0.03 43 Oct7AvgN TRUE 2.58E-08 36.6% 0.25 0.10 0.32 0.09 44 Oct7MaxN TRUE 1.44E-17 53.9% 1.48 0.41 2.05 0.48 45 Oct7Ratio TRUE 1.14E-08 37.4% 0.06 0.04 0.04 0.03 46 Oct7VarN TRUE 7.81E-13 45.9% 0.13 0.09 0.26 0.12
139
For each VCG signal recording, forty six features are calculated, but such a high-
dimensional feature space is not suitable for the classification in term of the
computational efficiency. Two sample KS tests between HC and MI help identify
important features that track the relevant cardiovascular state variations and are
insensitive to other extraneous variations, and determine an appropriate tight subset of
features which are most essential for the classification.
Table 8.2 shows that the Kolmogorov-Smirnov statistic variations of 46 features in
three groups. It may be noted that T_mag is the most significant feature in group I and II,
and its maximum difference level is 58%. We can find out eight features whose KSSTAT
(D) values are greater than 55%, and they are Oct0MaxN (69.3%), Oct0AvgN (67.8%),
Oct0VarN (64.7%), Oct2AvgN (61.7%), T_pos (61.6%), Oct2VarN (59.7%), T_mag
(58%) and Oct2MaxN (55.7%). VCG octant features in group III contribute seven out of
eight such important features. The statistical feature analysis implies that the eight feature
subsets incorporate most of the discriminating information of the total forty six features.
It is also intriguing to discover that octant 0, octant 2, and T vector related information
are the most prominent eight features to distinguish MIs from HCs, and they are used to
build the classification and regression tree (CART) model towards the detection of MI
and HC recordings in the Physionet PTB database.
8.4 Classification
The CART classification model uses decision rules learned from the pattern
(clustering) of features in a training data set to classify different groups of objects. Tree-
structured models are first proposed by Morgan and Songuist in 1963 for the analysis of
140
survey data [20]. This approach can be quite effective for creating clinical decision rules
that combine medical knowledgebase with data analysis.
Table 8.3 Experiment results of CART classification model using stochastic training datasets
Run No.Training data points
% Testing AccuracyMI HC Total
percentage MI (368) HC (80) Mean Std Mean Std Mean Std1 10.0% 37 8 92.34 4.50 77.47 15.96 89.70 4.492 20.0% 74 16 95.48 2.56 78.09 9.06 92.39 2.333 33.3% 123 27 95.97 2.25 80.68 9.73 93.26 1.764 50.0% 184 40 96.84 1.65 81.50 7.66 94.11 1.445 66.7% 246 54 96.40 1.98 83.54 7.67 94.16 1.826 75.0% 276 60 97.03 1.77 82.75 9.00 94.49 1.797 80.0% 295 64 96.89 2.29 82.38 8.63 94.30 2.218 90.0% 332 72 97.08 2.88 82.50 15.39 94.43 3.00
* Each run randomly selects the training data points in the specified percentage and does the classification for 100 times. The classification accuracy statistics are only calculated from testing dataset.
Table 8.3 presents the CART classification model experiment results for PTB
database with randomly selected percentage of training datasets. The training dataset size
is varied from 10% to 90% of the whole data. It may be noted that the average MI testing
accuracy is higher than 92.34% for all eight training sizes, but the standard deviation of
MI testing accuracy is a little bigger when the training data size is as small as 10% or as
large as 90%. This is resulted from the random selection of training datasets, and the
feature space may not be fully explored for a small portion of data. This small portion of
data, either in the testing or training group, may not be captured by the training samples’
patterns. The mean testing accuracy for HC is higher than 77.47%, and standard deviation
of HC testing accuracy follows the same trend as MI. While most HC testing standard
deviations are bigger than MI, this is because the HC population size (80) is less than MI
141
(368). This random training and testing experiment shows the generality and
effectiveness of CART classification model with vectorcardiographic octant features.
Figure 8.6 CART model cross validation curves and Tree size selection
To prevent the well-known “overfitting” problems in the machine learning model,
cross-validation techniques are utilized to find a generalized, optimal and simpler tree
structure for the MI and HC classification. In our case we remove a subset of 10% of the
data, build a tree using the other 90%, and use the tree to classify the removed 10%. This
process is repeated by removing each of ten subsets one at a time. As shown in Figure
8.6, the resubstitution error is kept decreasing as the tree size grows, but the cross-
validation results show that the error rate increases with the tree size beyond a certain
point. The best choice of pruned tree has the small cross-validation error and is roughly
as good as a more complex tree.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Number of terminal nodes
Cos
t (m
iscl
assi
ficat
ion
erro
r)Cross-validationResubstitutionMin + 1 std. err.Best choice
142
The optimal tree structure for PTB database MI and HC classification is as shown
in Figure 8.7. It may be noted that 96.74% MI and 96.25% HC in the PTB database can
be correctly detected with two simple if-then rules and four VCG octant features as
following:
(1) If Oct0MaxN < 0.52 or T_pos < 5.5, then the recording is MI;
(2) For the rest recordings after step 1, If Oct2AvgN < 0.123 or Oct0VarN > 0.015,
then the recording is MI;
Figure 8.7 Optimal CART tree structure for the classification of MI in PTB database
The detailed classifications of each step are as shown in Figure 8.8. The MI
recording is represented as red / in the feature space and HC as blue . In step 1, 24 MIs
and 4 HCs are incorrectly classified. For the overlapping area with 24 MIs and 76 HCs in
step 1, applying step 2 rules will correctly classify 14 MIs and 75 HCs. Therefore, the
detail accuracies can be calculated as following:
Pruned branches
Rule 2
Rule 1
143
Step 1: %MI = 100×(1-28/368) = 92.39%; %HC = 100×(1-2/80) = 97.50%
Step 2: %MI = 100×(1-12/368) = 96.74%; %HC = 100×(1-3/80) = 96.25%
(a)
(b) Figure 8.8 Detailed classification of PTB database MI and HC in (a) step 1 and (b) step 2
Figure 8.9 shows the healthy control (368) and myocardial infarction (80)
recordings in the PTB database characterized by three features. The first feature is
Oct0MaxN, which is the maximal vector magnitude in the first octant. The second feature
T position is the T wave peak’s octant location. The third feature Oct2AvgN is the
average vector magnitude in the octant 2. These three octant features discriminate the
Oct0MaxN
T Po
sitio
n
28 MI and2 HC misclassified
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
Oct2AvgN
Oct
0Var
N
12 MI and1 HC misclassified
0 0.05 0.1 0.15 0.2 0.250
0.005
0.01
0.015
0.02
0.025
144
healthy from myocardial infarction subjects. The healthy controls are shown to be cluster
in the green circle area, while myocardial infarction patients spread outside that green
area.
Figure 8.9 Healthy control (368) and myocardial infarction (80) distribution in 3D feature space (Oct0MaxN, T octant position and Oct2AvgN)
8.5 Discussions
It is generally agreed that Q wave, ST segment, and T wave are very substantial for
the identification of myocardial infarction. But conventional ECG intervals and
previously proposed VCG features are not adequate to capture the essential
characteristics of MI. The newly proposed VCG octant features can amazingly achieve
high MI and HC classification accuracies using two simple rules and four VCG octant
features. It may also be noted that step 1 can detect 92.39% of MI and 97.5% of HC with
just two features Oct0MaxN and T octant position.
HealthyCluster
145
(a)
(b)
(c)Figure 8.10 Oct0MaxN (a), T octant position (b) and Oct2AvgN (c) histogram plots
Figure 8.10 shows the distribution plots of feature Oct0MaxN, T octant position
and Oct2AvgN. It is may be noted that Oct0MaxN has significant distribution differences
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100Oct0MaxN
Magnitude
Freq
uenc
y
0 1 2 3 4 5 6 70
20
40
60
80
100
120T Wave Position
Octant
Freq
uenc
y
0 0.1 0.2 0.3 0.40
20
40
60
80Oct2AvgN
Magnitude
Freq
uenc
y
146
between MI (0.38±0.16) and HC (0.54±0.04). In addition, the majority (76/80) of HCs’ T
vector positions is also found to cluster in octant 6 (x-positive, y-positive, z-negative),
while MIs spread over all the eight octants. Q wave is frequently traveling around octant
0 in the VCG trajectory because Q wave is mostly negative in x, y, and z directions.
Oct2AvgN also shows dissimilarity in the distribution between healthy and myocardial
infarction patients. Both octant 0 and octant 2 are found to be located in the right anterior
side. Therefore, myocardial infarction subjects are statistically shown to have abnormal
electrical activities in the right anterior positions of the body. The VCG octant features
are shown to yield more statistical differences than conventional interval features in
Figure 8.5 and Table 8.2. The extra useful information can be found in 3D VCG spatial
dynamics than 12-lead ECG signals. It will also be much convenient to extract VCG
octant features with computer analysis than inspecting the 12-lead ECG traces and
memorizing ECG interval rules.
8.6 Summary
In this investigation, octant 0, octant 2, and T vector information in
vectorcardiogram are found to be very important for the diagnostics of heterogeneous
MIs. Statistical feature analysis shows that cardiac vector length distributions in octant 0
and octant 2, T vector length and octant position are the most prominent features to
distinguish MIs from HCs. It is significant at 69% level (p-value=9.63E-29) for the
Oct0MaxN (the maximum vector length in octant 0) distribution differences between MI
(0.38±0.16) and HC (0.54±0.04). The majority (76/80) of HCs’ T vector positions is also
found to cluster in octant 6 (x-positive, y-positive, z-negative), while MIs spread over all
the eight octants. Additionally, Oct2AvgN and Oct0VarN show >60% statistical
147
distribution differences. With only four octant features (Oct0MaxN, T_pos, Oct2AvgN
and Oct0VarN), a simple Classification and Regression Tree (CART) model can yield
classification accuracy with 96.74% sensitivity and 96.25% specificity. The stochastic
experiments with different percentage of training data size also reveal high sensitivity
(mean: 96%) and specificity (mean: 82%) for heterogeneous MI and HC classification,
which demonstrate the generality and effectiveness of CART model and
vectorcardiographic octant features. This study is definitely indicative of potential
clinical applications of MI diagnostic model from the proposed VCG octant features.
8.7 References
[1] D. Dubin, Rapid Interpretation of EKG's: An Interactive Course: Cover
Publishing Company, 2000.
[2] G. E. Dower and H. B. Machado, "XYZ data interpreted by a 12-lead computer
program using the derived electrocardiogram," Journal of electrocardiology, vol.
12, pp. 249-261, 1979.
[3] G. E. Dower, H. B. Machado, and J. A. Osborne, "On deriving the
electrocardiogram from vectorcardiographic leads," vol. 3, p. 87, 1980.
[4] G. E. Dower, A. Yakush, S. B. Nazzal, R. V. Jutzy, and C. E. Ruiz, "Deriving the
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G. Moody, C. Peng, and H. Stanley, "PhysioBank, PhysioToolkit, and PhysioNet:
Components of a New Research Resource for Complex Physiologic Signals,"
Circulation 101, vol. 23, pp. e215-e220, 2000 (June 13).
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[6] G. B. Moody, H. Koch, and U. Steinhoff, "The PhysioNet/Computers in
Cardiology Challenge 2006: QT interval measurement," in Computers in
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[7] F. Jager, A. Taddei, G. B. Moody, M. Emdin, G. Antolic, R. Dorn, A. Smrdel, C.
Marchesi, and R. G. Mark, "Long-term ST database: a reference for the
development and evaluation of automated ischaemia detectors and for the study of
the dynamics of myocardial ischaemia," Medical & Biological Engineering &
Computing, vol. 41, pp. 172-183, 2003.
[8] G. Bortolan and I. Christov, "Myocardial infarction and ischemia characterization
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633-636.
[9] D. Lemire, C. Pharand, J. Rajaonah, B. Dube, and A. R. LeBlanc, "Wavelet time
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ischemia," Heart Rhythm, vol. 1, pp. 317-325, Sep 2004.
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angle predicts cardiac death in a clinical population," Heart Rhythm, vol. 2, pp.
73-78, Jan 2005.
[12] C. D. Cowdery, G. S. Wagner, J. W. Starr, G. Rogers, and J. C. Greenfield, "New
Vectorcardiographic Criteria for Diagnosing Right Ventricular Hypertrophy in
Mitral-Stenosis - Comparison with Electrocardiographic Criteria," Circulation,
vol. 62, pp. 1026-1032, 1980.
[13] P. Hugenholtz, H. D. Levine, and G. H. Whipple, "A Clinical Appraisal of
Vectorcardiogram in Myocardial Infarction .1. Cube System," Circulation, vol.
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[14] T. C. Chou, Masangka.Mp, R. Young, G. F. Conway, and R. A. Helm, "Simple
Quantitative Vectorcardiographic Criteria for Diagnosis of Right Ventricular
Hypertrophy," Circulation, vol. 48, pp. 1262-1267, 1973.
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E. Boersma, "The electrical T-axis and the spatial QRS-T angle are independent
predictors of long-term mortality in patients admitted with acute ischemic chest
pain," Cardiology, vol. 101, pp. 199-207, 2004.
[16] H. Yang, M. Malshe, S. T. S. Bukkapatnam, and R. Komanduri, "Recurrence
quantification analysis and principal components in the detection of myocardial
infarction from vectorcardiogram signals," in Proceedings of the 3rd INFORMS
Workshop on Data Mining and Health Informatics (DM-HI), Washington, DC,
2008.
[17] W. Carson and Y. Z. Tseng, "Maximal Spatial St-Vector Patterns in Patients with
Acute Anteroseptal Myocardial-Infarction," International Journal of Cardiology,
vol. 43, pp. 165-173, Feb 1994.
[18] G. E. Burch, J. A. Abildskov, and J. A. Cronvich, "Studies of the Spatial
Vectorcardiogram in Normal Man," Circulation, vol. 7, pp. 558-572, 1953.
[19] A. Uehata, A. Kurita, B. Takase, K. Mizuno, K. Isozima, K. Satomura, K.
Arakawa, T. Shibuya, N. Aosaki, F. Osuzu, K. Hosono, H. Hayakawa, K.
Munakawa, and S. Seki, "Incidences of Coronary-Artery Stenosis and Mean Qrs
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1963.
PART III
SIMULATION AND PROGNOSTIC MODELING
150
CHAPTER IX
LOCAL RECURRENCE PREDICTION IN NONSTATIONARY CHAOTIC
SYSTEMS
This chapter reports a local recurrence modeling approach for state and
performance predictions in complex nonlinear and nonstationary systems.
Nonstationarity is treated as the switching force between different stationary systems,
which is shown as a series of finite time detours of system dynamics from the vicinity of
an attractor of a nonlinear process. Recurrence characteristics of the attractor are used to
partition the system trajectory into multiple near-stationary segments. Consequently,
piece-wise eigen analysis of ensembles in each near-stationary segment can capture both
nonlinear stochastic dynamics and nonstationarities. Extensive studies using simulated
and real world datasets reveal significant prediction accuracy improvements over other
alternative methods.
9.1 Introduction
Complex dynamics of many real-world systems can be effectively captured using
nonlinear stochastic dynamic models of the form:
( ) ( )dx F x dt x d�
(Eq 9-1)
where x is a m-dimensional state vector, F(•) is usually a nonlinear vector field, and the
stochastic term (x)d� accounts for the influence of extraneous phenomena. Nonetheless,
151
in many real life situations, only a few process output measurements y, or coupled high
dimensional measurements are available in lieu of the complete state vector x.
In such scenarios, traditional system identification and time series prediction
approaches are widely used to model the process outputs y as a linear or nonlinear
function of p past values of y and q previous realizations of independent noise (shock)
events #t. ARMA (autoregressive moving average) models assume both stationarity and
linearity, while ARIMA models incorporate a simple case of nonstationarity, namely low
frequency components [1]. These classical time series models cannot capture the
complexities presented in nonlinear chaotic processes. Prediction based on Kalman filters
tend to heavily depend on the model structure [2]. Polynomial models with global or
arbitrary local support tend to be unstable and not adequate for extrapolation [3]. Neural
network and sequential Bayesian models are known to need prohibitively large datasets
for prediction under highly nonstationary conditions and they consume significant model
training times [4]. Delay differential equations 1( ) ( ( ), ( ), , ( )n n n n kx t f x t x t x t� � � �� �
work well under deterministic conditions, but they pose acute stability issues due to
possible trajectory crossing under noisy conditions. Remarkably, none of the previous
approaches are generally suitable for prediction under nonlinear and highly nonstationary
conditions. Most of these conventional linear and nonlinear prediction approaches
assume stationarity and model system dynamics by deterministic parts plus random
measurement noise. The predictable deterministic parts often get distorted to some extent
by time domain differencing, smoothing or frequency domain filtering. In particular,
complex systems generate high-dimensional, nonlinear, nonstationary, noisy datasets
with dozens to hundreds (to even thousands) of state variables. The presence of data
152
complexity and high level nonstationarities significantly deteriorate the traditional
methods’ near-term prediction accuracies.
Meanwhile, modern industries are investing in a variety of sensors and plant floor
information systems to carefully monitor and manage the engineering systems including
automotive assembly lines, power plants, microelectronic manufacturing system, and
financial markets, etc. Large amounts of datasets and cheaper computational resources
offer an unprecedented opportunity to predict manufacturing system state and
performance from a nonlinear dynamic systems theoretic standpoint. Nonlinear dynamic
approaches can provide a better understanding of underlying physical mechanisms, and
give an advantage for the prediction of system behaviors.
The present approach exploits the inherent nonlinear stochastic dynamics of
complex manufacturing systems to improve predictive capability under noisy and
nonstationary conditions. Here, nonstationarity includes not just simple drifts in various
statistical moments over time, but also intermittent low and high dimensional chaotic
behaviors resulting from the random fluctuations of the model parameters. Central to our
approach is the segmentation of global performance signature (time-series) into multiple
stationary segments using recurrence analysis [5]. Stationary segmentation will lead to
reduced order models that can capture the local evolution patterns (including the
complexity and nonstationarities) better than any global model.
The remainder of this chapter is organized as follows: Section 9.2 reviews the
relevant work of state space prediction and recurrence analysis; Research methodology is
presented in Section 9.3; Section 9.4 demonstrates the simulation and real world
153
experiments of local recurrence prediction model; Section 9.5 concludes the reported
research.
9.2 Background
Nonlinear dynamic prediction approaches rely on reconstructing the state space of a
nonlinear process from the measured signals y(t). Takens’ embedding theorem [6] stated
as follows is the theoretic base for the complex system state space reconstruction.
Takens’ delay embedding theorem (1981): Let M be a compact manifold of topological
dimension d. For pairs ( 4 , h), where 4 : M M is a smooth (at least C2)
diffeomorphism and h: M R a smooth function, it is a generic property that the (2d+1)
- fold observation map Hk [4 , h]: M R2d+1 defined by x | (h(x), h(4 (x)), . . . , h(42d(x))) is an immersion (i.e. Hk is one-to-one between M and its image with both Hk and
Hk�1 differentiable).
Figure 9.1 Graphical illustration of the relationship among Lorenz time series, attractor and recurrence plot
0 500 1000 1500 2000-20
0
20y
Area V
x(500) x(800)
D(500,800)
x(500)
x(800)
Area U
x(500) = [y(500), y(500+�), y(500+2�)]
x(800) = [y(800), y(800+�), y(800+2�)]
y(800) y(500) (a)
(b)
(c)
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Takens’ delay embedding theorem shows the system dynamics can be adequately
reconstructed by using the time-delay coordinates of the individual measurements
because of the high dynamic coupling existing in physical system. For discrete time
series y (see Figure 9.1 (a)), state vector x (Figure 9.1 (b)) is reconstructed using a delay
sequence of points in {y(tn)} as,
x(tn) = [y(tn), y(tn+�), y(tn+2�), ..., y(tn+(m-1)�)] (Eq 9-2)
where m is the embedding dimension (m�2d+1) and � is the time delay. The optimal
sufficient embedding dimension m to unfold the attractor is determined by false nearest
neighbor method [7]. The mutual information function [8] is used to minimize both
nonlinear dynamic and linear correlations for the choice of optimal time delay �.
System dynamics often manifest in the vicinity of an attractor A (e.g., Lorenz
attractor [9] shown in Figure 9.1 (b)), which is an invariant set defined in an m-
dimensional state space. Lorenz attractors show unique topological patterns where
trajectory evolution switches between two cycle areas (area U and area V). State space
prediction approaches look for the relationships of the present system state x (a sequence
of points in the measured time series) to the nearby states in the embedding space and
their evolutions. It has been shown that global nonlinear dynamics can be synthesized by
multiple linear components, each of which shares the same local evolutions in an
attractor’s vicinity [10]. The digressions of local evolution patterns, specifically, finite
time detours from an attractor’s vicinity, are treated as nonstationarities. The
aforementioned nonstationarity may result from chaos, linear model switching or
155
parameter fluctuations (dynamic noise) and/or many forms of transient and intermittent
behaviors.
Despite high levels of nonlinearity, stochasticity and nonstationarity in the complex
system behaviors, local evolution of system state space often exhibits recurrence
characteristics as stated in the following theorem.
Poincare recurrence theorem (1890) [11]: Let T be a measure-preserving
transformation of a probability space (A, �), and let 5A be a measurable set. Then for
any J% � , �({x% |{Tj(x)}j�J5A\ })=0.
Poincare recurrence theorem implies that if one has a measure preserving
transformation, the trajectories eventually reappear at neighborhood 6 of former points
with probability one. A recurrence plot (see Figure 9.1 (c)) is used to identify the
aforementioned nonlinear and nonstationary structures in the state space. It delineates the
distances of every point x(ti), the state vector realized at time ti, to all the others in the
reconstructed state space, i.e., ( , ) : ( ( ) ( ) )i j i jD t t x t x t 1 � , where ||·|| is a distance
measurement (e.g., the Euclidean norm) and 1 (·) is the color code that maps the distance
to a color scale [12]. As shown in Figure 9.1, the distances between the 500th and 800th
points in the state space is shown as a color code at the coordinates (500,800) and (800,
500) of the recurrence plot. If the color code at the recurrence plot is blue, then the points
are located close to each other in the state space, and if the color code is red, the points
are located farther apart. Thus, a recurrence plot represents the topological relationships
existing in the m-dimensional state space in the form of 2D images. The ridges locate the
nonstationarities and the switchings between local behaviors [13], for e.g., the system
156
evolves from one behavior (area U) to the other (area V) (see Figure 9.1). The
separations between dark diagonal lines (along 45o degree) indicate the time periods
between the recurring system behaviors over certain time segments. Recurrence-based
methods have of late shown potential for representation and de-noising of measurements
from complex systems [14, 15]. This present work is one of the first investigations into
the use of recurrence analysis for state space prediction in highly nonstationary
conditions.
9.3 Local attractor topology analysis
The present local recurrence modeling approach is built on a premise, heretofore
unexplored, that despite high levels of nonstationarity, system dynamics are structurally
stable and nonstationarities are treated to cause finite time detours from the attractor’s
vicinity. Therefore, local recurrence characteristics, exhibited over certain time segments,
are leveraged to delineate the various local evolution patterns and nonstationary regimes
(finite-time detours) from attractor A and develop compact local models.
As summarized in Figure 9.2, central to our approach is the partitioning of a
measured time series into multiple near-stationary segments. Every strand of the system
trajectory within each segment bears a similar evolution pattern. Therefore stationary
assumptions hold therein within each segment. Pattern analysis and segmentation are
performed on a recurrence plot D(ti, tj) to identify these time segments. This segmentation
leads to reduced order models that can effectively capture the local evolutions of dynamic
system’s state space (including complexity and nonstationarity) better than with any
stationary model. For illustration purposes, let us consider a Lorenz attractor shown in
Figure 9.1 (b). All trajectories emerging from a region (e.g., U) have a similar local
157
evolution pattern. However, these local evolution patterns can be vastly dissimilar for the
trajectories emerging from two arbitrary regions, say, U and V. Local dynamics within
each of these regions can be simplified as unstable periodic orbits (UPOs) defined about
certain fixed points, and the global dynamics can be derived from concatenating the
trajectories from each of these regions as stated in the following.
Figure 9.2 Overall framework of local recurrence prediction model
Proposition 1: Dynamics about an attractor of complex nonlinear system can be
approximated by linear affine systems linked by switching laws, i.e.,
� �( ) ( )ii i i
ix F x f s x A x x7 8 � �� � ���
(Eq 9-3)
158
where Ai defines the local linear dynamics of affine subsystems, s(•) defines the switching
surface, fi[•] is a Boolean switching function that transits the dynamics among multiple
linear or affine systems, each defined over fixed points xi, and �i determine the locations
of the switching surface [10].
It has recently been shown that the system trajectories within a region i locally
evolve as unstable periodic orbits defined about certain saddle or foci type fixed points xi
[10]. Such local dynamics for region i can be captured at a specified parameter setting
using a linear system of the form ( ) .( )iiix A x x �� .
The system trajectories in an attractor can be derived from solving this piecewise
affine model. The global trajectories switch between the multiple local patterns as they
evolve about saddle type fixed points, i.e.,
(1) (2) ( )Zx x x x 9 9 9� . For instance, a
Lorenz attractor may be approximated using piecewise affine model consisting of two
linear components. The Jacobians of the respective linear/affine part are evaluated about
the corresponding saddles or foci type fixed points [10]. The saddle type fixed points help
determine the boundaries of the orbits defined about the foci.
Switching from one piecewise affine system to another takes place whenever the
system trajectories shift from exhibiting an unstable periodic behavior about a fixed point
to the other. The Boolean functions fi[•] is defined as following:
1, ( ) 0[ ( ) ]
0, ( ) 0i
i ii
s xf s x
s x7
77
� $+� - � �. (Eq 9-4)
This function is designed to make sure that only one of them equals unity at a given
location �i. A very minor perturbation in the states can switch the evolution path from one
region to the other. The points forming the transition states are rare or far from the local
regions U and V, i.e., 1{ ( )}Bi ix b : ; ;U V� . Within regions U and V, the trajectory flow
159
lines are laminar, and the divergence rate between closeby trajectories is close to zero.
The transition points are marked by significant changes in the divergence rates.
Recurrence patterns can be used to systematically identify the transitions. A
recurrence plot is symmetric about the diagonal, and patterns are equally distributed
along horizontal and vertical directions. If the distribution of states x(t) in the embedding
space is homogeneous, then the recurrence plot is also homogeneous. The presence of
such homogenous recurrence patterns implies a typically stationary process which has
short relaxation times in comparison with time spanned by recurrence plot. Interruptions
in the homogenous patterns indicate that some states over these times {x(bi)}Bi=1 are not
interior points of local sets U or V. Those states correspond to transition points between
various local homogeneous sets [13]. Hence, the disruptions in the recurrence plot can be
used to detect finite time detours from local vicinity of attractors, namely
nonstationarities.
The transition points between various local homogeneous sets, resulting from the
underlying chaotic dynamics and/or nonstationarities, can be detected by applying the
Sobel operator Iu and Iv on recurrence plot D(ti, tj). Where Iu and Iv are defined in terms of
a 3×3 matrix pair
1 0 12 0 21 0 1
�� �� � �� �� � � �
uI
1 2 10 0 01 2 1
� �� � � �� �� � � �
vI (Eq 9-5)
The intensity gradient magnitude Gu,v at a point (ti, tj) in the recurrence plot is given by
2 2, u v u vG G G , and Gu and Gv are the horizontal and vertical gradient estimations
defined as following:
*u uG I D and *v vG I D (Eq 9-6)
160
where * is the convolution or vector product operation.
Since Tu vI I and recurrence plot is symmetrical, any of the three gradient
measurements |Gu|, |Gv| or Gu,v can be equally used to extract the transition boundaries.
From equation (3), we can see that Sobel operators are equivalent to 2-D high pass filters.
It can compactly capture the high frequency nonhomogeneous patterns better than the
original recurrence plots.
Figure 9.3 Edge detection in a Lorenz recurrence plot
Thus, by applying the Sobel operator to recurrence plots, we reduce the amount of
data to be processed, filter out information that may be regarded as less relevant, and
161
preserve the important nonhomogeneous structural properties. Boundaries of linear and
stationary segments are shown as horizontal and vertical edges in the Gu(ti, tj) image.
Histogram similarities of adjacent sliding windows are compared to locate these
boundaries. Such edge detection methods give prominence to high frequency components
in the image, for e.g., the boundaries of objects, the boundaries of surface markings or
curves that correspond to discontinuities in surface orientation. The length of each
segment depends on the local divergence rate of trajectories emanating from a given
neighborhood. As shown in Figure 9.3, the transition states or saddle points {x(bi)}Bi=1 are
identified in the Lorenz recurrence plot (also refer to Figure 9.1). The enlarged portion in
Figure 9.3 shows the homogeneity of point distributions in each segment.
1ix i
Kx
Nx
( )ix b
2Nx 1Nx
3Nx 4Nx 5Nx
Figure 9.4 Local eigen prediction model from ensembles within a stationary segment
In our approach, ensembles are generated within a near-stationary segment to
provide a more homogeneous set for eigen analysis. Therefore eigen representation can
capture the essential dynamics without being influenced by nonstationarities in the state
space. The local eigen representation of segment i uses the K ensembles extracted from
the nearest neighbors {xik}K
k=1 of a segment’s starting point x(bi) (see Figure 9.4). The
ensembles of length Li are collected from the historical data such that K+Li < N, the
162
length of time series. All the ensembles are generated from the realized states, and
consequently, the model is causal. Intuitively, each eigenfunction estimated from those
ensembles �( ) captures the shape of a specific mode of variation (roughly, a degree of
freedom) within the segment [16]. The leading variation modes of a system with d active
degrees of freedom can be obtained as the projection of the ensembles onto the space
spanned by �1( ), �2( ), …, �d( ). Thus, the local eigen representation captures major
variations of the ensembles as well as nuances of the ensembles in each stationary
segment, and it is extrapolated along the leading eigen directions to predict the future
states (see Figure 9.4). Whenever the system evolves (due to chaos) or drifts (due to
nonstationarity) into a different regime, a new local eigen representation is established to
capture the new patterns. The advantage of local representation is evident from the
following result.
Proposition 2: Local eigen (L2-optimal) representations obtained from segmenting a
chaotic attractor are more parsimonious than a global ensemble representation.
More often than not, the operators Ai defined in equation (1) either do not have a
full rank, or have a highly ill-conditioned eigen system where one or more eigen values
are significantly lower than the rest. For example, in affine approximation of Lorenz-like
systems [10], the local eigenvectors of Ai’s tend to be significantly smaller along the
directions normal to the planes of the “butterfly wings,” and generally speaking, along
the so-called nullclinic directions [10], of a Lorenz attractor compared to the other
orthogonal directions. The eigen directions that lie along the nullclinic directions, and
hence the signal components projected along these directions, can be ignored. Therefore
much of the salient evolutions about a certain fixed point on the attractor can be captured
163
by considering very few eigen basis functions. Consequently, the local representations,
defined between two successive “switching points,” will be more compact than a
representation based on global ensembles. From a noise reduction standpoint,
consideration of fewer basis functions implies that signal energy is concentrated along
limited number of directions.
9.4 Simulation experiment results
In this section, we report our evaluation of the performance of local recurrence
model using both simulated and real world datasets. The prediction results are compared
to traditional time series models (e.g. ARMA) and some of the well known nonlinear
models (e.g. polynomial, neural network, radial basis function (RBF)). It may be noted
that few if any of the previous approaches is suitable for prediction under highly
nonstationary and noisy conditions. Polynomial models and delay differential equation
models are not included for comparison because the experimental results are observed to
easily diverge from or fail to predict the actual system behaviors. Only radial basis
function models provided some promising predictability. However the local recurrence
model is able to separate the similar evolution patterns of chaotic systems under highly
nonstationary conditions.
The following two scenarios are simulated towards evaluating the local recurrence
modeling approach: (1) nonlinear and stationary signals generated by contaminating the
first component of Lorenz attractor [9] with three different noise levels (signal noise
ratios (SNR) between 7.37dB-21.35dB); (2) nonlinear and nonstationary signals obtained
by dividing the contaminated Lorenz time series into several segments and randomly
rearranging the sequence of these segments (SNR varied between 7.37dB-21.35dB).
164
(a)
(b)
Figure 9.5 (a) top - original Lorenz time series, middle – random Gaussian noise, bottom – random permutated Lorenz time series contaminated by noise; (b) reconstructed noise-
free Lorenz attractor vs. random permutated Lorenz attractor contaminated by noise
Figure 9.5 (a) depicts the comparisons between original Lorenz time series and the
randomly permutated noisy Lorenz time series. As illustrated in the Figure 9.5 (a) bottom
plot, the structures of noise-free Lorenz time series are demolished by random noise and
artificial generated nonstationarity. Figure 9.5 (b) evidently shows that it is difficult to
-20
0
20
-20
0
20
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-20
0
20
165
identify the clean patterns in the random permutated and noise contaminated Lorenz
attractor. The nonstationarity and noise in the simulated data pose additional issues for
the state space prediction. Local recurrence model delineates the near-stationary
segments and suppresses noise components to track the relevant variations underlying the
process and not the extraneous phenomena. The prediction results are compared at
multiple locations to those from RBF and two classical time series models: ARMA(1,0) -
a prediction model currently used in the practice, and ARMA(3,3) - the “optimal” model
form emerged from a classical system identification standpoint.
Figure 9.6 shows the comparisons of prediction error statistics for noisy Lorenz
time series (signal range: -25.25~27.25) in scenario (1). It may be noted that ARMA(1,0)
step one prediction errors vary from 2.48~7.38 under three noise levels, ARMA(3,3)
from 1.75~7.06, and local recurrence model from 1.93~6.25. Here, the aforementioned
prediction error statistics are calculated in terms of root mean square errors between
actual ix and predicted value ix� at N prediction locations. In noise level 1, step one
prediction errors are relevant low with reference to the signal range, and comparable for
ARMA and local recurrence models. However, with the increase of noise levels, ARMA
prediction models become inferior to the local recurrence model. For instance, the local
recurrence model’s one step prediction RMS errors are lower by 12% (around 0.81~1.13)
than both ARMA models in noise level 3 (see Figure 9.6). In addition, ARMA models
become increasingly worse (around 0.75~5.06 higher RMS errors) than local recurrence
model for multiple step-ahead (�2) predictions under all three noise levels. This is
because linear models cannot capture the chaotic dynamics underlying the Lorenz time
series.
166
Figure 9.6 Five step prediction RMS error comparisons for Lorenz time series under different noise levels with various prediction models (ARMA(1,0), ARMA(3,3), RBF
model and local recurrence model)
Both RBF and the local recurrence model can capture the system nonlinearity, but
RBF model does not consider nonstationarity and is very sensitive to stochastic
conditions. The prediction errors from the RBF models progressively deteriorate with an
167
increase of noise levels, for e.g., step one prediction errors are elevated from 1.33 (noise
level 1) to 7.44 (noise level 3). The deterioration rate of RBF model is found to be higher
than local recurrence model which is from 1.93 (noise level 1) to 6.25(noise level 3). In
noise level 3, local recurrence model five step prediction RMS errors vary from 4.17 to
6.25, on the other hand, RBF is from 7.23 to 8.74 (see Figure 9.6). The local recurrence
model is shown to perform better under highly noisy conditions. It may also be noted that
the prediction errors of the local recurrence model do not increase with prediction steps,
for example, ARMA(1,0) prediction errors are increased from 4.20 to 8.45 with
prediction steps in noise level 2, while variations of the local recurrence model (between
3.60 and 4.81) do not follow with prediction steps (see Figure 9.6). This is because eigen
analysis of ensembles in a near-stationary segment guarantees that predictions always lie
within the bounds of the reconstructed attractor. In contrast, RBF models depend on the
size of training datasets. Multiple step-ahead predictions can sustain excessive divergence
and instability because of the stochastic conditions and/or accumulated iterated prediction
errors. As shown in noise level 2 in Figure 9.6, the RBF model is unstable and prediction
results diverge with the steps. In a nutshell, Figure 9.6 shows that the local recurrence
model can capture the nonlinearity as well as stochasticity in this case.
168
Figure 9.7 Five step prediction RMS error comparisons for randomly permutated Lorenz time series under different noise levels with various prediction models (ARMA(1,0),
ARMA(3,3), RBF model and local recurrence model)
Figure 9.7 shows the prediction error statistics in scenario (2) – randomly
permutated noisy Lorenz time series. The random permutation in arbitrarily lengths will
169
generate finite time detours from the vicinity of an attractor. This type of nonstationarity
is quite different from the commonly considered seasonal trends. In this experiment, the
local recurrence model is shown to significantly outperform all the other three models
under all three noise levels. For five step prediction RMS errors under three noise levels,
ARMA(1,0) ranges from 4.21 to 10.62, ARMA(3,3) from 3.85 to 10.64, RBF from 4.96
to 52.07, and the local recurrence model from 3.81 to 7.12. Moreover, noise level 1 plot
(see Figure 9.7) reveals that nonstationarity increases the probability to drive RBF model
unstable and divergent for more than one step predictions.
9.5 Summary
The present work is one of the first attempts to use recurrence analysis for
prediction in the presence of generic forms of nonstationarities. It combines the principles
of statistical estimation with dynamic systems theory. Most of current applications
threshold recurrence plots for the exploration of nearest neighbors
( , ) : ( ( ) ( ) )i j i jD t t x t x t r 1 � �� . In contrast, the proposed approach exploits the
topological irregularity information contained in recurrence plots, and uses edge
detection and pattern matching methods for stationary segmentation. The transition states
that occur when the system behaviors are switched have found to be effectively captured.
Experimentations using simulated datasets reveal the superiority of local recurrence
model over other alternative models for nonlinear stochastic systems under transient
conditions. It is conceivable that such a method can be applied to improve predictability
of similar complex physical systems under nonstationary conditions.
170
9.6 References
[1] R. H. Shumway and D. S. Stoffer, Time Series Analysis and Its Applications: With
R Examples: Springer, 2006.
[2] L. Ljung, System Identification - Theory For the User. N.J: PTR Prentice Hall,
1999.
[3] M. Small, K. Judd, and A. Mees, "Modeling continuous processes from data,"
Physical Review E, vol. 65, p. 046704, 2002.
[4] A. Freking, W. Kinzel, and I. Kanter, "Learning and predicting time series by
neural networks," Physical Review E, vol. 65, p. 050903, 2002.
[5] S. T. S. Bukkapatnam, S. R. Kumara, and A. Lakhtakia, "Local eigenfunctions-
based suboptimal wavelet packet representation of contaminated chaotic signals,"
IMA Journal of Applied Mathematics, vol. 63, pp. 149-160, 1999.
[6] F. Takens, Detecting strange attractors in turbulence, In Lecture notes in
mathematics vol. 898: Springer, Berlin, 1981.
[7] M. B. Kennel, R. Brown, and H. D. I. Abarbanel, "Determining embedding
dimension for phase-space reconstruction using a geometrical construction,"
Physical Review A, vol. 45, p. 3403, 1992.
[8] A. M. Fraser and H. L. Swinney, "Independent coordinates for strange attractors
from mutual information," Physical Review A, vol. 33, p. 1134, 1986.
[9] E. N. Lorenz, "Atmospheric predictability as revealed by naturally occurring
analogues," Journal of Atmosphere Sciences, vol. 26, pp. 636-646, 1969.
[10] F. V. A. Gleison, L. Christophe, and A. Luis Antonio, "Piecewise affine models
of chaotic attractors: The Rossler and Lorenz systems," Chaos: An
Interdisciplinary Journal of Nonlinear Science, vol. 16, p. 013115, 2006.
171
[11] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical
Systems, 1 ed.: Cambridge University Press, 1995.
[12] J.-P. Eckmann, S. O. Kamphorst, and D. Ruelle, "Recurrence plots of dynamical
systems," Europhys Lett, vol. 4, pp. 973-977, 1987.
[13] N. Marwan, M. Carmen Romano, M. Thiel, and J. Kurths, "Recurrence plots for
the analysis of complex systems," Physics Reports, vol. 438, pp. 237-329, 2007.
[14] L. Matassini, H. Kantz, J. Holyst, and R. Hegger, "Optimizing of recurrence plots
for noise reduction," Physical Review E, vol. 65, p. 021102, 2002.
[15] R. Hegger, H. Kantz, and L. Matassini, "Denoising Human Speech Signals Using
Chaoslike Features," Physical Review Letters, vol. 84, p. 3197, 2000.
[16] H. Yang, S. T. Bukkapatnam, and R. Komanduri, "Nonlinear adaptive wavelet
analysis of electrocardiogram signals," Physical Review E (Statistical, Nonlinear,
and Soft Matter Physics), vol. 76, p. 026214, 2007.
172
CHAPTER X
RADIAL BASIS FUNCTION SIMULATION MODELING OF VCG SIGNALS
This chapter will introduce an approach to construct a reasonable data-driven model
for the analysis of heart system using heart monitoring signals (VCG and ECG). Such a
heart model can help us gain a deeper understanding of underlying heart dynamics and
explore the relationship among those signals. Moreover this data-driven model can be
used to simulate the heart electric activities under a specific disease, and it will greatly
facilitate the further research of cardiac diseases.
Figure 10.1 Overall framework of RBF simulation model of VCG signals
As shown in Figure 10.1, this chapter presents an approach to model and simulate the
vectorcardiogram (VCG) and standard 12-lead electrocardiogram (ECG) using radial
basis functions uniformly distributed along the VCG trajectory. In the first step, the
EKG Analyzer
Poincare sectioning
Vectorcardiogram
Ensemble generation
Pattern analysis
VCG –> 12 lead transform
RBF VCG model estimation
Simulation & model diagnostic
Pattern
Model
173
ensembles will be extracted by Poincare sectioning the VCG trajectory, and they will be
aligned to the interval [0 2�] for the pattern analysis. Then we use the least square fit
procedure to fit a Radial Basis Function (RBF) network structure to the extracted
ensembles. The well trained RBF network model will provide the closely approximate of
the real VCG data and share the similar 3D VCG trajectories. The standard 12 lead ECG
signals are derived from VCG by transformation, and the transformation matrix can
either use the famous Dower matrix [1] or be estimated from the experiment data [2-4].
Finally, the simulation model is evaluated and compared to the real experiment data in
the time domain, frequency domain, and the recurrence quantification plots.
10.1 Poincare sectioning of vectorcardiogram
Figure 10.2 (a) An illustration of trajectories of an attractor intersecting a (planar) Poincare section; (b) Poincare sectioning of VCG trajectory
Poincare section is a dE-1 dimensional hyperplane intersecting with the phase space
trajectories (see Figure 10.2 (a)). The recurrence property of a chaotic attractor A shows
that for every and almost every A such that , in
effect, the trajectories with an attractor remain bounded. Those points , i =1, 2, … at
which the trajectory intersects the Poincare section follow a return map. For strictly
0# � (0)x % , 0t& (0) ( )x x t #� �
iP
(a) (b) Poincare section �
Trajectory
174
periodic trajectory, the points , i =1, 2, …will overlap (i.e., ��0) such that the duration
between to along the trajectory constitutes the period.
Figure 10.3 X, Y and Z ensemble components extracted from the Poincare sectioning of VCG trajectory
For near-periodic signals, such as VCG, each strand emanating from a Poincare
section intersection point Pi and lasting approximately till the next intersection Pi+1 along
the trajectory may be treated as a realization of a stochastic process from an invariant
probability space [5] (see Figure 10.2 (b)). Due to heart rate variability, some ensembles
move faster, i.e., the two successive intersections occur over shorter intervals, compared
to the others [6]. In this chapter, all the extracted ensembles will be aligned in the time
interval [0 2�] (see
Figure 10.3), and the heartbeat variations will be simulated as a Weibull
distribution as shown in the following section 10.2.
10.2 Heart rate variability
The variations of RR interval in the ECG/VCG time series produce the well-known
heart rate variability. It has long been understood that a metronomic heart rate is
iP
iP 1iP
175
pathological, and that the healthy heart is influenced by multiple neural and hormonal
inputs that result in variations in interbeat (RR) intervals, at time scales ranging from less
than a second to 24 hours [7]. There is evidence that the underlying heart beat dynamics
may have a multi-fractal temporal structure [8]. Heart rate variability analysis has been
shown to provide an effective evaluation of cardiac disease such as different stages in the
atrial fibrillation [9].
Figure 10.4 Weibull distribution fitting of RR interval time series
The complex interdependencies of heart beat variations at different scales make it
difficult to model the RR interval time series, and each recording in the PTB database has
less than120 heart beats. Therefore, this chapter will only simulate the heartbeat
variations as a Weibull distribution.
The Weibull distribution can mimic the behavior of other statistical distributions
such as the normal and the exponential [10]. The Probability Density Function PDF
associated with the Weibull distribution is:
,<
,=>
,.
,-+
���
����
���
�
��
����
���
� bb xxbxf(((
exp)(1
(Eq 10-1)
And the associated Cumulative Distribution Function (CDF) is a stretched exponential:
176
���
�
��
����
�����
bxxF(
exp1)( (Eq 10-2)
Where b is the Weibull slope, also referred to as the “shape parameter”, and ( is the
characteristic value, also referred to as the “scale parameter”. The Weibull distribution is
fitted into the heartbeat time series extracted from the Poincare sectioning of
vectorcardiogram trajectories (see Figure 10.4), and it was found to yield better
distribution evaluation than normal and uniform distribution. The estimated parameters b
and ( will be used for the simulation of RR interval in the later section.
10.3 Research methodology
Figure 10.5 Uniform distributed radial basis function centers along VCG trajectory
The modeling of vectorcardiogram is that given N time vectors in ?1: t1, ..., tN, and
their corresponding target values in ?3: y1, ..., yN, we are seeking a projection h:?1@ ?3,
so that A1)i)N, h(ti) = yi using the radial basis function network. The basis functions are
using Gaussian functions, and the number of basis functions may be selected as N for the
exact interpolation [11, 12]. However, this might lead to poor and complex results for
new vectors by overfitting the data. For a compact solution with good generalization, the
number of basis functions used in the VCG model is reduced from N to M (M << N) (see
177
Figure 10.5). The training vectorcardiogram (VCG) data is from the ensembles extracted
from the Poincare sectioning.
The output of the VCG RBF network model is given by
( )j j
jy w t0 � (Eq 10-3)
2( ) exp( 2 )j jt t c0 B � � (Eq 10-4)
where c is the centre of the radial basis function j, t is the input data, and ||t –cj|| is the
measure of the distance of the point from the radial centre, also called the Euclidean
distance. From Eq 10-4, we can make the basis matrix G for all the centres (j = 1, 2, …,
M), weight matrix W and target matrix VN as following:
C D1 1 1 2 1
2 1 2 2 2
1 2
( ) ( ) ( )( ) ( ) ( )
0 2
( ) ( ) ( )
N
N
M M M N
t t tt t t
G M N t
t t t
0 0 00 0 0
E
0 0 0
� �� �� � 2 %� �� � �
��
� � � ��
(Eq 10-5)
11 12 1
22 22 2
3 3 3
3
M
M
M M M
w w ww w w
W M
w w w
� �� �� � 2� �� � �
� � � (Eq 10-6)
1 2
1 2
1 2
3x x xN
N y y yN
z z zN
v v vV v v v N
v v v
� �� � 2� �� � �
���
(Eq 10-7)
The N-sample estimate of the weight matrix NW� of the RBF network VCG model can be
determined to minimize the objective function ( , )N NF W V
� �
1
1( , )N
N N Nt
F W V V WGN
�� (Eq 10-8)
F Garg min ( , )N N Nw
W F W V� (Eq 10-9)
and the estimate can be obtained using the pseudoinverse:
178
� � 1T TNW V G G G
�
(Eq 10-10)
10.4 Transformation from VCG to standard 12 lead ECG signals
Numerous attempts have been made previously to estimate the transformation
matrix between 12-lead ECG and VCG signals [2, 13-17]. Dower matrix (see Eq 10-11)
is the most famous and generalized one, and it can be used to derive the 12-lead ECG
signals from the VCG. Therefore the whole measurements including 12-lead ECG and
VCG can be simulated by this investigated RBF model.
Dower transformation matrix
0.515 0.157 0.9170.044 0.164 1.3870.882 0.098 1.2771.213 0.127 0.6011.125 0.127 0.0860.831 0.076 0.2300.632 0.235 0.0590.235 1.066 0.132
D
� �� �� ��� �� ��� ��� � � ��� �� �� ��� �
�� � �
(Eq 10-11)
If we have the VCG data in matrix V (3×N), the eight leads (I, II, v1-v6) S (8×N)can be
derived by S=DV except the mathematical calculated augment leads.
This generalized Dower matrix can be used to derive 12-lead ECGs when we only
have VCG signals. If the 12-lead ECG signals are recorded simultaneously with VCG,
the transformation matrix S can also be estimated for the specified person using either the
linear or nonlinear regression (see Eq 10-12).
� � 1T TD S V V V�
� (Eq 10-12)
10.5 Implementation results
Figure 10.6, Figure 10.7, Figure 10.8, Figure 10.9, and Figure 10.10 show the
implementations results of radial basis simulation models for healthy control subjects,
179
and patients with myocardial infarction, valvular heart disease, or bundle branch block.
The simulated heart electrical activities in Frank XYZ orthogonal directions are
compared in time domain, frequency domain, and state space domain. It is found that the
investigated RBF simulation model can effective track the pathological morphology
patterns.
180
05
1015
2025
3035
400
100
200
Vx
Act
ual V
CG v
s. S
imul
ated
VCG
in F
requ
ency
Dom
ain
05
1015
2025
3035
400
200
Vy
05
1015
2025
3035
400
100
200
Vz
Freq
uenc
y (H
Z)
010
0020
0030
0040
0050
0060
0070
00
-0.20
0.2
Vx
Actu
al V
ecto
rcar
diog
ram
Sig
nal
010
0020
0030
0040
0050
0060
0070
00
-0.20
0.2
Vy
010
0020
0030
0040
0050
0060
0070
00-0
.200.
20.
4
Vz
1000
2000
3000
4000
5000
6000
7000
-0.20
0.2
Vx
Sim
ulat
ed V
ecto
rcar
diog
ram
Sig
nal
1000
2000
3000
4000
5000
6000
7000
-0.20
0.2
Vy
1000
2000
3000
4000
5000
6000
7000
-0.20
0.2
0.4
Vz
Cas
e 1:
Pat
ient
001/
s001
0_re
, Myo
card
ial i
nfar
ctio
n, in
fero
-late
ra
Figu
re 1
0.6
Patie
nt00
1/s0
010_
re (a
)Sim
ulat
ed V
CG
traj
ecto
ry v
s. ac
tual
V
CG
traj
ecto
ry; (
b) F
requ
ency
dom
ain
com
paris
on b
etw
een
actu
al V
CG
an
d si
mul
ated
VC
G; (
c) T
ime
dom
ain
com
paris
on b
etw
een
actu
al V
CG
and
si
mul
ated
VC
G
181
Cas
e 2:
Pat
ient
255/
s049
1_re
, hea
lthy
cont
rol
Figu
re 1
0.7
Patie
nt25
5/s0
491_
re (a
)Sim
ulat
ed V
CG
traj
ecto
ry v
s. ac
tual
VC
G tr
ajec
tory
; (b)
Fre
quen
cy d
omai
n co
mpa
rison
bet
wee
n ac
tual
VC
G a
nd si
mul
ated
VC
G; (
c) T
ime
dom
ain
com
paris
on
betw
een
actu
al V
CG
and
sim
ulat
ed V
CG
182
Cas
e 3:
Pat
ient
005/
s010
1lre
, Myo
card
ial I
nfar
ctio
n, A
nter
ior
Figu
re 1
0.8
Patie
nt00
5/s0
101l
re (a
)Sim
ulat
ed V
CG
traj
ecto
ry v
s. ac
tual
V
CG
traj
ecto
ry; (
b) F
requ
ency
dom
ain
com
paris
on b
etw
een
actu
al V
CG
an
d si
mul
ated
VC
G; (
c) T
ime
dom
ain
com
paris
on b
etw
een
actu
al V
CG
an
d si
mul
ated
VC
G
183
Cas
e 4:
Pat
ient
106/
s003
0_re
, Val
vula
r he
art d
isea
se, p
rost
hetic
val
ve
Figu
re 1
0.9
Patie
nt10
6/s0
030_
re (a
)Sim
ulat
ed V
CG
traj
ecto
ry v
s. ac
tual
V
CG
traj
ecto
ry; (
b) F
requ
ency
dom
ain
com
paris
on b
etw
een
actu
al V
CG
an
d si
mul
ated
VC
G; (
c) T
ime
dom
ain
com
paris
on b
etw
een
actu
al V
CG
an
d si
mul
ated
VC
G
184
Cas
e 5:
Pat
ient
175/
s000
9_re
, bun
dle
bran
ch b
lock
Figu
re 1
0.10
Pat
ient
175/
s000
9_re
(a)S
imul
ated
VC
G tr
ajec
tory
vs.
actu
al V
CG
traj
ecto
ry; (
b) F
requ
ency
dom
ain
com
paris
on b
etw
een
actu
al
VC
G a
nd si
mul
ated
VC
G; (
c) T
ime
dom
ain
com
paris
on b
etw
een
actu
al
VC
G a
nd si
mul
ated
VC
G
185
10.6 Summary
In this chapter, we have shown that the data-driven RBF network VCG model can
effectively capture the topology of actual VCG trajectory, and yield similar spectrum in
the frequency domain. The accurate modeling of VCG can facilitate the understanding of
the heart dynamics. The possible applications of this data-driven model are listed as
following:
(1) Feature extraction
The model parameters such as RBF weights, HRV distribution parameters can be
used as features for the further diagnosis. Therefore, numerous VCG and ECG
data is reduced to limited amount of features while preserve the same information.
(2) VCG and ECG data compression
We knew that hundreds of gigabytes VCG and ECG data will be stored for the
real-time high sampling ECG recorder. Since the RBF model replicates the actual
VCG topology, those parameters can be saved and used for simulating a long
duration VCG data in the future. Therefore, the VCG and ECG data are
compressed to save the disk storage space.
(3) Algorithm evaluation
This data-driven model can be fitted to different group of diseases and patients.
The well trained model for different pathologies will be able to generate several
various simulating VCG and ECG data which can be used to test the QRST
characteristic point detection algorithms, QRST cancellation algorithms, adaptive
filtering algorithms, and classification algorithms etc.
186
(4) Disease prognosis
The well trained model captures all the characteristics from the actual data, and
the actual real-time heart monitoring signals can be compared the simulation
model trained in the healthy conditions. The similarity differences can serve as
the performance measure for the prognostic purpose.
The investigated RBF VCG model compressed information about the actual VCG
trajectory dynamics, and the VCG signals can also be projected to standard 12-lead ECG
signals with the transformation matrix. The future research will focus on exploring the
above applications of the RBF network VCG model.
10.7 Reference
[1] G. E. Dower and H. B. Machado, "XYZ data interpreted by a 12-lead computer
program using the derived electrocardiogram," Journal of electrocardiology, vol.
12, pp. 249-261, 1979.
[2] D. E. Gustafson, A. Akant, and T. L. Johnson, "Linear transformations relating
the electrocardiogram and vectorcardiogram," Computers and Biomedical
Research, vol. 9, pp. 337-348, 1976.
[3] J. B. Kadtke and M. N. Kremliovsky, "Signal and pattern detection or
classification by estimation of continuous dynamical models." vol. 20020133317
United States: Nonlinear Solutions, Inc., 2002.
[4] D. M. Schreck, M. S. Viscito, and C. Brotea, "The derived N-Lead
electrocardiogram," Annals of Emergency Medicine, vol. 44, pp. S73-S74, 2004.
[5] H. Yang, S. T. Bukkapatnam, and R. Komanduri, "Nonlinear adaptive wavelet
analysis of electrocardiogram signals," Physical Review E (Statistical, Nonlinear,
and Soft Matter Physics), vol. 76, p. 026214, 2007.
187
[6] F. Marciano, M. L. Migaux, D. Acanfora, G. Furgi, and F. Rengo, "Quantification
of Poincare maps for the evaluation of heart rate variability," 1994, pp. 577-580.
[7] T. Kuusela, T. Shepherd, and J. Hietarinta, "Stochastic model for heart-rate
fluctuations," Physical Review E, vol. 67, p. 061904, 2003.
[8] P. C. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z.
R. Struzik, and H. E. Stanley, "Multifractality in human heartbeat dynamics,"
Nature, vol. 399, pp. 461-465, 1999.
[9] S. T. S. Bukkapatnam, H. Yang, and R. Komanduri, "A comparative analysis of
alternative approaches for the classification of atrial fibrillation (AF) episodes
from EKG signals," Journal of Electrocardiography (under review).
[10] A. C. Cohen, "Maximum Likelihood Estimation in the Weibull Distribution
Based on Complete and on Censored Samples," Technometrics, vol. 7, pp. 579-
588, 1965.
[11] M. T. Hagan, H. B. Demuth, and M. H. Beale, Neural Network Design: PWS Pub.
Co., December 29, 1995.
[12] P. E. McSharry, G. D. Clifford, L. Tarassenko, and L. A. Smith, "A dynamical
model for generating synthetic electrocardiogram signals," IEEE Transactions on
Biomedical Engineering, vol. 50, pp. 289-294, 2003.
[13] S. P. Nelwan, J. A. Kors, S. H. Meij, J. H. van Bemmel, and M. L. Simoons,
"Reconstruction of the 12-lead electrocardiogram from reduced lead sets,"
Journal of electrocardiology, vol. 37, pp. 11-18, 2004.
[14] G. E. Dower, H. B. Machado, and J. A. Osborne, "On deriving the
electrocardiogram from vectorcardiographic leads," vol. 3, p. 87, 1980.
[15] G. E. Dower, A. Yakush, S. B. Nazzal, R. V. Jutzy, and C. E. Ruiz, "Deriving the
12-lead electrocardiogram from four (EASI) electrodes," Journal of
electrocardiology, vol. 21, pp. S182-S187, 1988.
188
[16] L. Edenbrandt and O. Pahlm, "Vectorcardiogram synthesized from a 12-lead
ECG: Superiority of the inverse Dower matrix," Journal of electrocardiology, vol.
21, pp. 361-367, 1988.
[17] D. Wei, T. Kojima, T. Nakayama, and Y. Sakai, "Method for deriving standard
12-lead electrocardiogram, and monitoring apparatus using the same." vol.
20020045837: Kohden Corporation, 2002.
189
CHAPTER XI
CONCLUSIONS AND FUTURE WORK
One hundred years ago, researchers were striving on the design of meticulous
cardiovascular sensor monitoring system. Nowadays, a variety of sensor networks and
dedicated super computers are taken advantage to collect high precision heart monitoring
signals in real time. The huge amount of observation data offers an unprecedented
opportunity to understand the hidden heart nature buried in the sea of data. This
dissertation focused on data driven nonlinear stochastic modeling and analysis of
complex cardiovascular systems towards the diagnostic and prognostic applications. The
major conclusions and future work recommendations are as follows:
11.1 Conclusions
A novel customized wavelet function approach is developed to resolve the big data
issue in the cardiovascular monitoring field. This method uses near periodic patterns of
electrocardiogram (ECG) signals to optimally design a wavelet for the specific
cardiovascular applications. This customized wavelet successfully reduces the
compactness of ECG signal representation by around two orders of magnitude, which
directly increase data compression ratio and benefit the telemedicine.
Sophisticated visualization tools are also developed to explore unexpected patterns
and interpret hidden evidences. Traditional representation habits, like time domain
190
representation, can prevent us from extracting new understandings. Frequency domain,
time-frequency domain, animations, heart and torso finite element models, and state
space domain will give new promising insights of system dynamics from different
perspectives.
Advanced rigorous wavelet analysis and fractal dynamic study are also conducted
to quantitatively extract more sensitive features to cardiovascular pathological variations
instead of the extraneous noises. The medical diagnostic models are designed to
mathematically associate the effective nonlinear features with two studied cardiovascular
diseases - Myocardial Infarction (MI) and Atrial Fibrillation (AF). The classification
accuracies are found to be as high as 96% in sensitivity for MI and >90% for different
stages of AF.
In order to address the increasing complexity of datasets, a local recurrence model
is developed to make effective forecasting in the nonstationary chaotic environments.
This model investigates the system local recurrence behaviors by dividing the delay
reconstructed system state space into multiple stationary segments. Simulation study
shows significant prediction accuracy improvements over commonly used stationary
methods. This model is also successfully implemented to predict shift-wise throughputs
in multistage manufacturing assembly lines, and yields >74% accuracy improvement in
the real world.
191
11.2 Future work
Aiming at healthcare excellence, future work will keep on sensor based modeling
and analysis of complex heart system. The continued improvements and prospected
studies will be as follows:
First, risk prognostic frameworks for the complex cardiovascular system will be
further advanced and studied for the ischemia conditions and sleep apnea subjects. It is
conceptually known that human heart is a complex organ system with a built-in capacity
for self-regulation and adaptation. Myocardial infarction patients always have self-
organizing transitions between ischemic and non-ischemic conditions, and sleep apnea
patients have discontinuous pauses in respirations during sleep. There are high risks of
sudden cardiac death from these cardiovascular transitions. The research objective is to
on-line estimate the potential risk and real-time predict the future cardiovascular statuses.
The accurate predictions will not only have great academic and economic impacts, but
also give the cardiovascular patients a better chance for a healthy life.
In this dissertation, 12-lead ECG and 3-lead VCG signals are studied. The
investigation can be extended to multi-sensor-fusion and monitoring schemes of
cardiovascular systems. Monitoring sensor signals may include heart sound, respiration,
blood pressure, electrooculogram (EOG), electromyography (EMG),
electroencephalogram (EEG), magnetic resonance imaging (MRI), or nuclear magnetic
resonance imaging (NMRI) etc. Multi-sensor-fusion may also be combined with
nonlinear dynamics, fractal analysis, higher-order statistical processing and knowledge-
based approaches towards effective interpretations of physiological activities. Given the
192
complexity of underlying process, this research direction has high potentials to give rise
to the novel advanced signal processing approaches.
Thirdly, mathematical modeling of biomechanics in the physiological system will
facilitate not only the understanding of exhibited physiological signals, but also
autonomic organism functioning. The intriguing example is ancient Chinese medicine -
acupuncture and moxibustion. The stimulation of certain spots in human body will help
relieve pain and reach the body homeostasis. This acupuncture medicine is developed
through practice for thousands of years and shows effective applications in various kinds
of diseases like trigeminal neuralgia, palsy, tracheitis, arthritis and so on, but the
underlying mechanics have not been fully understood yet. Sensor based modeling
approach will assist the explanation of underlying biomechanics from a data-driven
perspective of view. The effective mathematical biomechanical modeling will help
further improve the current medicine treatments, and thus lead to the healthcare
excellence.
VITA
Hui Yang
Candidate for the Degree of
Doctor of Philosophy
Dissertation: NONLINEAR STOCHASTIC MODELING AND ANALYSIS OF CARDIOVASCULAR SYSTEM DYNAMICS – DIAGNOSTIC AND PROGNOSTIC APPLICATIONS
Major Field: Industrial Engineering and Management
Biographical:
Personal Data: Born in Nanjing, Jiangsu province, People’s Republic of China
Education: Received Bachelor of Science degree in Electrical Engineering from China University of Mining and Technology in Beijing, P. R. China, in June 2002; received Master of Science degree with a major in Electrical Engineering at China University of Mining and Technology in Beijing, P. R. China, in June 2005; completed Doctor of Philosophy degree with a major in Industrial Engineering and Management at Oklahoma State University in May, 2009.
Experience:Employed by Oklahoma State University, School of Industrial and Management as a graduate research assistant and teaching assistant, from August 2005 to December 2008.
Professional Memberships: � Institute for Operations Research and the Management Sciences (INFORMS) � The Industrial Engineering Honor Society (Alpha Pi Mu) � Institute of Electrical and Electronics Engineers (IEEE) � Institute for Industrial Engineers (IIE)� American Society of Quality (ASQ)
ADVISER’S APPROVAL: Dr. Satish T. S. Bukkapatnam Dr. Ranga Komanduri
Name: Hui Yang Date of Degree: May, 2009
Institution: Oklahoma State University Location: Stillwater, Oklahoma
Title of Study: NONLINEAR STOCHASTIC MODELING AND ANALYSIS OF CARDIOVASCULAR SYSTEM DYNAMICS – DIAGNOSTIC AND PROGNOSTIC APPLICATIONS
Pages in Study: 192 Candidate for the Degree of Doctor of Philosophy
Major Field: Industrial Engineering and Management
Scope and Method of Study:
The purpose of this investigation is to develop monitoring, diagnostic and prognostic schemes for cardiovascular diseases by studying the nonlinear stochastic dynamics underlying complex heart system. The employment of a nonlinear stochastic analysis combined with wavelet representations can extract effective cardiovascular features, which will be more sensitive to the pathological dynamics instead of the extraneous noises. While conventional statistical and linear systemic approaches have limitations for capturing signal variations resulting from changes in the cardiovascular system state. The research methodology includes signal representation using optimal wavelet function design, feature extraction using nonlinear recurrence analysis, and local recurrence modeling for state prediction.
Findings and Conclusions:
� New representation scheme for the ECG signals by an optimal wavelet function design which yields one order of magnitude compactness reduction compared to the conventional wavelets in terms of entropy.
� New spatiotemporal representation scheme for the VCG signals, which significantly assists the interpretation of vectorcardiogram and facilitates the exploration of related important spatiotemporal cardiovascular dynamics.
� Developed a classification and regression tree (CART) model to integrate the cardiologists’ knowledge towards accurate classification of Atrial Fibrillation (AF) states from sparse datasets.
� Developed a Myocardial Infarction (MI) classification model using recurrence quantification analysis and spatial octant features of VCG attractor dynamics, which detects MI up to about 96% accuracy in both sensitivity and specificity.
� Designed a local recurrence prediction model for real time medical prognosis and risk assessment of cardiovascular problems, which is particularly suitable for the implementation in the nonlinear and nonstationary environment.